Math Boot Camp - Class #5

Alex Vickery

Royal Holloway - University of London

28th September, 2017

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 1 / 40

Outline:Today’s Class

Summation Notation:

Rules for Sums:

Double Sums

Logic

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 2 / 40

Summation Notation:

Outline:Today’s Class

Summation Notation:

Rules for Sums:

Double Sums

Logic

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 3 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

Economists often use census data. Suppose a country is divided into sixregions.

Let Ni denote the population of region i , then the total population is givenby:

N1 + N2 + N3 + N4 + N5 + N6

It is convenient to have an abbreviated notation for such lengthy sums.

The capital Greek letter Σ is conventionally used as a summationsymbol, and the sum is written as:

6∑i=1

Ni

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 4 / 40

Summation Notation:

Summation Notation:Introduction:

The sum reads as “the sum from i = 1 to i = 6 of Ni”.

If there are n regions, then:

N1 + N2 + · · ·+ Nn

is one possible notation for the total population.

Here, the dots · · · indicate that the obvious pattern continues, but comesto an end before the last term Nn.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 5 / 40

Summation Notation:

Summation Notation:Introduction:

The sum reads as “the sum from i = 1 to i = 6 of Ni”.

If there are n regions, then:

N1 + N2 + · · ·+ Nn

is one possible notation for the total population.

Here, the dots · · · indicate that the obvious pattern continues, but comesto an end before the last term Nn.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 5 / 40

Summation Notation:

Summation Notation:Introduction:

The sum reads as “the sum from i = 1 to i = 6 of Ni”.

If there are n regions, then:

N1 + N2 + · · ·+ Nn

is one possible notation for the total population.

Here, the dots · · · indicate that the obvious pattern continues, but comesto an end before the last term Nn.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 5 / 40

Summation Notation:

Summation Notation:Introduction:

The sum reads as “the sum from i = 1 to i = 6 of Ni”.

If there are n regions, then:

N1 + N2 + · · ·+ Nn

is one possible notation for the total population.

Here, the dots · · · indicate that the obvious pattern continues, but comesto an end before the last term Nn.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 5 / 40

Summation Notation:

Summation Notation:Introduction:

The sum reads as “the sum from i = 1 to i = 6 of Ni”.

If there are n regions, then:

N1 + N2 + · · ·+ Nn

is one possible notation for the total population.

Here, the dots · · · indicate that the obvious pattern continues, but comesto an end before the last term Nn.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 5 / 40

Summation Notation:

Summation Notation:Introduction:

In summation or sigma notation, we use the summation symbol Σ andwrite:

n∑i=1

Ni

This tells us to form the sum of all terms that result when we substitutesuccessive integers for i , starting with i = 1 and ending with i = n.

The symbol i is called the index of summation.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 6 / 40

Summation Notation:

Summation Notation:Introduction:

In summation or sigma notation, we use the summation symbol Σ andwrite:

n∑i=1

Ni

This tells us to form the sum of all terms that result when we substitutesuccessive integers for i , starting with i = 1 and ending with i = n.

The symbol i is called the index of summation.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 6 / 40

Summation Notation:

Summation Notation:Introduction:

In summation or sigma notation, we use the summation symbol Σ andwrite:

n∑i=1

Ni

This tells us to form the sum of all terms that result when we substitutesuccessive integers for i , starting with i = 1 and ending with i = n.

The symbol i is called the index of summation.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 6 / 40

Summation Notation:

Summation Notation:Introduction:

In summation or sigma notation, we use the summation symbol Σ andwrite:

n∑i=1

Ni

This tells us to form the sum of all terms that result when we substitutesuccessive integers for i , starting with i = 1 and ending with i = n.

The symbol i is called the index of summation.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 6 / 40

Summation Notation:

Summation Notation:Introduction:

The upper and lower limits of summation can both vary. For example:

35∑i=30

Ni = N30 + N31 + N32 + N33 + N34 + N35

is the population in the six regions numbered from 30 to 35.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 7 / 40

Summation Notation:

Summation Notation:Introduction:

The upper and lower limits of summation can both vary. For example:

35∑i=30

Ni = N30 + N31 + N32 + N33 + N34 + N35

is the population in the six regions numbered from 30 to 35.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 7 / 40

Summation Notation:

Summation Notation:Introduction:

The upper and lower limits of summation can both vary. For example:

35∑i=30

Ni = N30 + N31 + N32 + N33 + N34 + N35

is the population in the six regions numbered from 30 to 35.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 7 / 40

Summation Notation:

Summation Notation:Introduction:

More generally, suppose p and q are integers with q ≥ p. Then:

q∑i=p

ai = ap + ap+1 + · · ·+ aq

denotes the sum that results when we substitute successive integers for i ,starting with i = p and ending with i = q.

If the upper and lower limits of the summation are the same, then the“sum” reduces to one term.

If the upper limit is less than the lower limit, then the sum reduces to zero.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 8 / 40

Summation Notation:

Summation Notation:Introduction:

More generally, suppose p and q are integers with q ≥ p. Then:

q∑i=p

ai = ap + ap+1 + · · ·+ aq

denotes the sum that results when we substitute successive integers for i ,starting with i = p and ending with i = q.

If the upper and lower limits of the summation are the same, then the“sum” reduces to one term.

If the upper limit is less than the lower limit, then the sum reduces to zero.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 8 / 40

Summation Notation:

Summation Notation:Introduction:

More generally, suppose p and q are integers with q ≥ p. Then:

q∑i=p

ai = ap + ap+1 + · · ·+ aq

denotes the sum that results when we substitute successive integers for i ,starting with i = p and ending with i = q.

If the upper and lower limits of the summation are the same, then the“sum” reduces to one term.

If the upper limit is less than the lower limit, then the sum reduces to zero.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 8 / 40

Summation Notation:

Summation Notation:Introduction:

More generally, suppose p and q are integers with q ≥ p. Then:

q∑i=p

ai = ap + ap+1 + · · ·+ aq

denotes the sum that results when we substitute successive integers for i ,starting with i = p and ending with i = q.

If the upper and lower limits of the summation are the same, then the“sum” reduces to one term.

If the upper limit is less than the lower limit, then the sum reduces to zero.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 8 / 40

Summation Notation:

Summation Notation:Introduction:

More generally, suppose p and q are integers with q ≥ p. Then:

q∑i=p

ai = ap + ap+1 + · · ·+ aq

denotes the sum that results when we substitute successive integers for i ,starting with i = p and ending with i = q.

If the upper and lower limits of the summation are the same, then the“sum” reduces to one term.

If the upper limit is less than the lower limit, then the sum reduces to zero.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 8 / 40

Summation Notation:

Summation Notation:Introduction:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 9 / 40

Summation Notation:

Summation Notation:Introduction:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 10 / 40

Rules for Sums:

Outline:Today’s Class

Summation Notation:

Rules for Sums:

Double Sums

Logic

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 11 / 40

Rules for Sums:

Rules for Sums:Introduction:

The following properties of the sigma notation are helpful whenmanipulating sums:

n∑i=1

(ai + bi ) =n∑

i=1

ai +n∑

i=1

bi (additivity property)

n∑i=1

cai = cn∑

i=1

ai (homogeneity property)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 12 / 40

Rules for Sums:

Rules for Sums:Introduction:

The following properties of the sigma notation are helpful whenmanipulating sums:

n∑i=1

(ai + bi ) =n∑

i=1

ai +n∑

i=1

bi (additivity property)

n∑i=1

cai = cn∑

i=1

ai (homogeneity property)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 12 / 40

Rules for Sums:

Rules for Sums:Introduction:

The following properties of the sigma notation are helpful whenmanipulating sums:

n∑i=1

(ai + bi ) =n∑

i=1

ai +n∑

i=1

bi (additivity property)

n∑i=1

cai = cn∑

i=1

ai (homogeneity property)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 12 / 40

Rules for Sums:

Rules for Sums:Introduction:

The proofs are straightforward, for example:

n∑i=1

aci = ca1 + ca2 + · · ·+ can = c(a1 + a2 + · · ·+ an) = cn∑

i=1

ai

The homogeneity property states that a constant factor can be movedoutside the summation sign.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 13 / 40

Rules for Sums:

Rules for Sums:Introduction:

The proofs are straightforward, for example:

n∑i=1

aci = ca1 + ca2 + · · ·+ can = c(a1 + a2 + · · ·+ an) = cn∑

i=1

ai

The homogeneity property states that a constant factor can be movedoutside the summation sign.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 13 / 40

Rules for Sums:

Rules for Sums:Introduction:

The proofs are straightforward, for example:

n∑i=1

aci = ca1 + ca2 + · · ·+ can = c(a1 + a2 + · · ·+ an) = cn∑

i=1

ai

The homogeneity property states that a constant factor can be movedoutside the summation sign.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 13 / 40

Rules for Sums:

Rules for Sums:Introduction:

In particular, if ai = 1 for all i , then:

n∑i=1

c = nc

Which just states that a constant c summed n times is equal to n times c .

The summation rules can be applied in combination to give formulas like:

n∑i=1

(ai + bi − 2ci + d) =n∑

i=1

ai +n∑

i=1

bi − 2n∑

i=1

ci + nd

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 14 / 40

Rules for Sums:

Rules for Sums:Introduction:

In particular, if ai = 1 for all i , then:

n∑i=1

c = nc

Which just states that a constant c summed n times is equal to n times c .

The summation rules can be applied in combination to give formulas like:

n∑i=1

(ai + bi − 2ci + d) =n∑

i=1

ai +n∑

i=1

bi − 2n∑

i=1

ci + nd

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 14 / 40

Rules for Sums:

Rules for Sums:Introduction:

In particular, if ai = 1 for all i , then:

n∑i=1

c = nc

Which just states that a constant c summed n times is equal to n times c .

The summation rules can be applied in combination to give formulas like:

n∑i=1

(ai + bi − 2ci + d) =n∑

i=1

ai +n∑

i=1

bi − 2n∑

i=1

ci + nd

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 14 / 40

Rules for Sums:

Rules for Sums:Introduction:

In particular, if ai = 1 for all i , then:

n∑i=1

c = nc

Which just states that a constant c summed n times is equal to n times c .

The summation rules can be applied in combination to give formulas like:

n∑i=1

(ai + bi − 2ci + d) =n∑

i=1

ai +n∑

i=1

bi − 2n∑

i=1

ci + nd

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 14 / 40

Rules for Sums:

Rules for Sums:Introduction:

In particular, if ai = 1 for all i , then:

n∑i=1

c = nc

Which just states that a constant c summed n times is equal to n times c .

The summation rules can be applied in combination to give formulas like:

n∑i=1

(ai + bi − 2ci + d) =n∑

i=1

ai +n∑

i=1

bi − 2n∑

i=1

ci + nd

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 14 / 40

Rules for Sums:

Rules for Sums:Introduction:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 15 / 40

Rules for Sums:

Rules for Sums:Introduction:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 16 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

We all know that (a + b)1 = a + b and (a + b)2 = a2 + 2ab + b2. By thesame logic we can find:

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

etc ...

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 17 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The corresponding formula for (a+ b)m, where m is any natural number is:

(a + b)m = am +

(m1

)am−1b + · · ·+

(m

m − 1

)abm−1 +

(mm

)bm

This formula involves the binomial coefficients

(mk

), which are defined,

for m = 1, 2, · · · and for k = 0, 1, 2, · · · ,m, by:(mk

)=

m(m − 1) · · · (m − k + 1)

k!,

(m0

)= 1

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 18 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The corresponding formula for (a+ b)m, where m is any natural number is:

(a + b)m = am +

(m1

)am−1b + · · ·+

(m

m − 1

)abm−1 +

(mm

)bm

This formula involves the binomial coefficients

(mk

), which are defined,

for m = 1, 2, · · · and for k = 0, 1, 2, · · · ,m, by:(mk

)=

m(m − 1) · · · (m − k + 1)

k!,

(m0

)= 1

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 18 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The corresponding formula for (a+ b)m, where m is any natural number is:

(a + b)m = am +

(m1

)am−1b + · · ·+

(m

m − 1

)abm−1 +

(mm

)bm

This formula involves the binomial coefficients

(mk

), which are defined,

for m = 1, 2, · · · and for k = 0, 1, 2, · · · ,m, by:

(mk

)=

m(m − 1) · · · (m − k + 1)

k!,

(m0

)= 1

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 18 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The corresponding formula for (a+ b)m, where m is any natural number is:

(a + b)m = am +

(m1

)am−1b + · · ·+

(m

m − 1

)abm−1 +

(mm

)bm

This formula involves the binomial coefficients

(mk

), which are defined,

for m = 1, 2, · · · and for k = 0, 1, 2, · · · ,m, by:(mk

)=

m(m − 1) · · · (m − k + 1)

k!,

(m0

)= 1

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 18 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

In general,

(m1

)= m and

(mm

)= 1. When m = 5 for example, we have:

(52

)=

5 · 41 · 2

,

(53

)=

5 · 4 · 31 · 2 · 3

,

(54

)=

5 · 4 · 3 · 21 · 2 · 3 · 4

= 5

Then (a + b)5 gives us:

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 19 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

In general,

(m1

)= m and

(mm

)= 1. When m = 5 for example, we have:

(52

)=

5 · 41 · 2

,

(53

)=

5 · 4 · 31 · 2 · 3

,

(54

)=

5 · 4 · 3 · 21 · 2 · 3 · 4

= 5

Then (a + b)5 gives us:

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 19 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

In general,

(m1

)= m and

(mm

)= 1. When m = 5 for example, we have:

(52

)=

5 · 41 · 2

,

(53

)=

5 · 4 · 31 · 2 · 3

,

(54

)=

5 · 4 · 3 · 21 · 2 · 3 · 4

= 5

Then (a + b)5 gives us:

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 19 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

In general,

(m1

)= m and

(mm

)= 1. When m = 5 for example, we have:

(52

)=

5 · 41 · 2

,

(53

)=

5 · 4 · 31 · 2 · 3

,

(54

)=

5 · 4 · 3 · 21 · 2 · 3 · 4

= 5

Then (a + b)5 gives us:

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 19 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 20 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The numbers in the triangle are indeed the binomial coefficients. Forinstance, the numbers in row 6 (given that the first is row 0) are:

(60

) (61

) (62

) (63

) (64

) (65

) (66

)Note that the numbers are symmetric about the middle line. This

symmetry can be expressed as:(mk

)=

(m

m − k

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 21 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The numbers in the triangle are indeed the binomial coefficients. Forinstance, the numbers in row 6 (given that the first is row 0) are:(

60

) (61

) (62

) (63

) (64

) (65

) (66

)

Note that the numbers are symmetric about the middle line. This

symmetry can be expressed as:(mk

)=

(m

m − k

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 21 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The numbers in the triangle are indeed the binomial coefficients. Forinstance, the numbers in row 6 (given that the first is row 0) are:(

60

) (61

) (62

) (63

) (64

) (65

) (66

)Note that the numbers are symmetric about the middle line. This

symmetry can be expressed as:

(mk

)=

(m

m − k

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 21 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

The numbers in the triangle are indeed the binomial coefficients. Forinstance, the numbers in row 6 (given that the first is row 0) are:(

60

) (61

) (62

) (63

) (64

) (65

) (66

)Note that the numbers are symmetric about the middle line. This

symmetry can be expressed as:(mk

)=

(m

m − k

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 21 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

Apart from the 1 at both ends of each row, each number is the sum of thetwo adjacent numbers in the row above.

For instance, 56 in the eighth row is equal to the sum of 21 and 35 in theseventh row.

In symbols: (m + 1k + 1

)=

(mk

)+

(m

k + 1

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 22 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

Apart from the 1 at both ends of each row, each number is the sum of thetwo adjacent numbers in the row above.

For instance, 56 in the eighth row is equal to the sum of 21 and 35 in theseventh row.

In symbols: (m + 1k + 1

)=

(mk

)+

(m

k + 1

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 22 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

Apart from the 1 at both ends of each row, each number is the sum of thetwo adjacent numbers in the row above.

For instance, 56 in the eighth row is equal to the sum of 21 and 35 in theseventh row.

In symbols:

(m + 1k + 1

)=

(mk

)+

(m

k + 1

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 22 / 40

Rules for Sums:

Rules for Sums:Newtons Binomial Formula:

Apart from the 1 at both ends of each row, each number is the sum of thetwo adjacent numbers in the row above.

For instance, 56 in the eighth row is equal to the sum of 21 and 35 in theseventh row.

In symbols: (m + 1k + 1

)=

(mk

)+

(m

k + 1

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 22 / 40

Double Sums

Outline:Today’s Class

Summation Notation:

Rules for Sums:

Double Sums

Logic

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 23 / 40

Double Sums

Double Sums:Introduction:

Often one has to combine several summation signs. Consider, for example,the following rectangular array of numbers:

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The array can be regarded as a spreadsheet.

A typical number in the array is of the form aij , where 1 ≤ i ≤ m and1 ≤ j ≤ n.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 24 / 40

Double Sums

Double Sums:Introduction:

Often one has to combine several summation signs. Consider, for example,the following rectangular array of numbers:

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The array can be regarded as a spreadsheet.

A typical number in the array is of the form aij , where 1 ≤ i ≤ m and1 ≤ j ≤ n.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 24 / 40

Double Sums

Double Sums:Introduction:

Often one has to combine several summation signs. Consider, for example,the following rectangular array of numbers:

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The array can be regarded as a spreadsheet.

A typical number in the array is of the form aij , where 1 ≤ i ≤ m and1 ≤ j ≤ n.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 24 / 40

Double Sums

Double Sums:Introduction:

Often one has to combine several summation signs. Consider, for example,the following rectangular array of numbers:

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The array can be regarded as a spreadsheet.

A typical number in the array is of the form aij , where 1 ≤ i ≤ m and1 ≤ j ≤ n.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 24 / 40

Double Sums

Double Sums:Introduction:

Let us find the sum of all the numbers in the array by first summing allnumbers in each of the m rows, then adding all the row sums.

The m different row sums can be written in the form:

n∑j=1

a1j ,n∑

j=1

a2j , · · · ,n∑

j=1

amj ,

The sum of these m sums is equal to:

n∑j=1

a1j +n∑

j=1

a2j + · · ·+n∑

j=1

amj =m∑i=1

( n∑j=1

aij

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 25 / 40

Double Sums

Double Sums:Introduction:

Let us find the sum of all the numbers in the array by first summing allnumbers in each of the m rows, then adding all the row sums.

The m different row sums can be written in the form:

n∑j=1

a1j ,n∑

j=1

a2j , · · · ,n∑

j=1

amj ,

The sum of these m sums is equal to:

n∑j=1

a1j +n∑

j=1

a2j + · · ·+n∑

j=1

amj =m∑i=1

( n∑j=1

aij

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 25 / 40

Double Sums

Double Sums:Introduction:

Let us find the sum of all the numbers in the array by first summing allnumbers in each of the m rows, then adding all the row sums.

The m different row sums can be written in the form:

n∑j=1

a1j ,n∑

j=1

a2j , · · · ,n∑

j=1

amj ,

The sum of these m sums is equal to:

n∑j=1

a1j +n∑

j=1

a2j + · · ·+n∑

j=1

amj =m∑i=1

( n∑j=1

aij

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 25 / 40

Double Sums

Double Sums:Introduction:

Let us find the sum of all the numbers in the array by first summing allnumbers in each of the m rows, then adding all the row sums.

The m different row sums can be written in the form:

n∑j=1

a1j ,n∑

j=1

a2j , · · · ,n∑

j=1

amj ,

The sum of these m sums is equal to:

n∑j=1

a1j +n∑

j=1

a2j + · · ·+n∑

j=1

amj =m∑i=1

( n∑j=1

aij

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 25 / 40

Double Sums

Double Sums:Introduction:

Let us find the sum of all the numbers in the array by first summing allnumbers in each of the m rows, then adding all the row sums.

The m different row sums can be written in the form:

n∑j=1

a1j ,n∑

j=1

a2j , · · · ,n∑

j=1

amj ,

The sum of these m sums is equal to:

n∑j=1

a1j +n∑

j=1

a2j + · · ·+n∑

j=1

amj =m∑i=1

( n∑j=1

aij

)

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 25 / 40

Double Sums

Double Sums:Introduction:

If instead we add the numbers in each of the n columns first and then addthese sums we get:

m∑i=1

ai1 +m∑i=1

ai2 + · · ·+m∑i=1

aim =n∑

j=1

( m∑i=1

aij

)In both cases, we have calculated the sum of all the numbers in the array.For this reason, we must have:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

This says that in a (finite) double sum, the order of summation isimmaterial.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 26 / 40

Double Sums

Double Sums:Introduction:

If instead we add the numbers in each of the n columns first and then addthese sums we get:

m∑i=1

ai1 +m∑i=1

ai2 + · · ·+m∑i=1

aim =n∑

j=1

( m∑i=1

aij

)

In both cases, we have calculated the sum of all the numbers in the array.For this reason, we must have:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

This says that in a (finite) double sum, the order of summation isimmaterial.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 26 / 40

Double Sums

Double Sums:Introduction:

If instead we add the numbers in each of the n columns first and then addthese sums we get:

m∑i=1

ai1 +m∑i=1

ai2 + · · ·+m∑i=1

aim =n∑

j=1

( m∑i=1

aij

)In both cases, we have calculated the sum of all the numbers in the array.For this reason, we must have:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

This says that in a (finite) double sum, the order of summation isimmaterial.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 26 / 40

Double Sums

Double Sums:Introduction:

If instead we add the numbers in each of the n columns first and then addthese sums we get:

m∑i=1

ai1 +m∑i=1

ai2 + · · ·+m∑i=1

aim =n∑

j=1

( m∑i=1

aij

)In both cases, we have calculated the sum of all the numbers in the array.For this reason, we must have:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

This says that in a (finite) double sum, the order of summation isimmaterial.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 26 / 40

Double Sums

Double Sums:Introduction:

If instead we add the numbers in each of the n columns first and then addthese sums we get:

m∑i=1

ai1 +m∑i=1

ai2 + · · ·+m∑i=1

aim =n∑

j=1

( m∑i=1

aij

)In both cases, we have calculated the sum of all the numbers in the array.For this reason, we must have:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

This says that in a (finite) double sum, the order of summation isimmaterial.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 26 / 40

Double Sums

Double Sums:Introduction:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 27 / 40

Logic

Outline:Today’s Class

Summation Notation:

Rules for Sums:

Double Sums

Logic

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 28 / 40

Logic

Logic:Propositions:

Assertions that are either true or false are called propositions.

“All individuals who breathe are alive” is an example of a true proposition.

“All individuals who breathe are healthy” is a flase one.

Note that if the words used to express such an assertion lack precisemeaning, it will often be difficult to tell whether or not the proposition istrue or false.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 29 / 40

Logic

Logic:Propositions:

Assertions that are either true or false are called propositions.

“All individuals who breathe are alive” is an example of a true proposition.

“All individuals who breathe are healthy” is a flase one.

Note that if the words used to express such an assertion lack precisemeaning, it will often be difficult to tell whether or not the proposition istrue or false.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 29 / 40

Logic

Logic:Propositions:

Assertions that are either true or false are called propositions.

“All individuals who breathe are alive” is an example of a true proposition.

“All individuals who breathe are healthy” is a flase one.

Note that if the words used to express such an assertion lack precisemeaning, it will often be difficult to tell whether or not the proposition istrue or false.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 29 / 40

Logic

Logic:Propositions:

Assertions that are either true or false are called propositions.

“All individuals who breathe are alive” is an example of a true proposition.

“All individuals who breathe are healthy” is a flase one.

Note that if the words used to express such an assertion lack precisemeaning, it will often be difficult to tell whether or not the proposition istrue or false.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 29 / 40

Logic

Logic:Implications:

In order to keep track of each step in a chain of logical reasoning, it oftenhelps to use implication arrows.

Suppose P and Q are two propositions such that whenever P is true, thenQ is necessarily true.

In this case, we usually write:

P =⇒ Q

This is read as “P implies Q, or “if P then Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 30 / 40

Logic

Logic:Implications:

In order to keep track of each step in a chain of logical reasoning, it oftenhelps to use implication arrows.

Suppose P and Q are two propositions such that whenever P is true, thenQ is necessarily true.

In this case, we usually write:

P =⇒ Q

This is read as “P implies Q, or “if P then Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 30 / 40

Logic

Logic:Implications:

In order to keep track of each step in a chain of logical reasoning, it oftenhelps to use implication arrows.

Suppose P and Q are two propositions such that whenever P is true, thenQ is necessarily true.

In this case, we usually write:

P =⇒ Q

This is read as “P implies Q, or “if P then Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 30 / 40

Logic

Logic:Implications:

In order to keep track of each step in a chain of logical reasoning, it oftenhelps to use implication arrows.

Suppose P and Q are two propositions such that whenever P is true, thenQ is necessarily true.

In this case, we usually write:

P =⇒ Q

This is read as “P implies Q, or “if P then Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 30 / 40

Logic

Logic:Implications:

In order to keep track of each step in a chain of logical reasoning, it oftenhelps to use implication arrows.

Suppose P and Q are two propositions such that whenever P is true, thenQ is necessarily true.

In this case, we usually write:

P =⇒ Q

This is read as “P implies Q, or “if P then Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 30 / 40

Logic

Logic:Implications:

Other ways of expressing the same implication include:

“Q if P”, “P only if Q”.

The symbol =⇒ is an implication arrow, and it points in the direction ofthe logical implication.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 31 / 40

Logic

Logic:Implications:

Other ways of expressing the same implication include:

“Q if P”, “P only if Q”.

The symbol =⇒ is an implication arrow, and it points in the direction ofthe logical implication.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 31 / 40

Logic

Logic:Implications:

Other ways of expressing the same implication include:

“Q if P”, “P only if Q”.

The symbol =⇒ is an implication arrow, and it points in the direction ofthe logical implication.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 31 / 40

Logic

Logic:Implications:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 32 / 40

Logic

Logic:Implications:

In certain cases where the implication is valid, it may be possible to draw alogical conclusion in the other direction:

Q =⇒ P

In such cases, we can write both implications together in a single logicalequivalence:

P ⇐⇒ Q

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 33 / 40

Logic

Logic:Implications:

In certain cases where the implication is valid, it may be possible to draw alogical conclusion in the other direction:

Q =⇒ P

In such cases, we can write both implications together in a single logicalequivalence:

P ⇐⇒ Q

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 33 / 40

Logic

Logic:Implications:

In certain cases where the implication is valid, it may be possible to draw alogical conclusion in the other direction:

Q =⇒ P

In such cases, we can write both implications together in a single logicalequivalence:

P ⇐⇒ Q

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 33 / 40

Logic

Logic:Implications:

In certain cases where the implication is valid, it may be possible to draw alogical conclusion in the other direction:

Q =⇒ P

In such cases, we can write both implications together in a single logicalequivalence:

P ⇐⇒ Q

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 33 / 40

Logic

Logic:Implications:

We then say that “P is equivalent to Q”.

because we have both “P if Q” and “P only if Q”, we also say “P if andonly if Q”.

We often write “P iff Q” for short.

The symbol ⇐⇒ is an equivalence arrow.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 34 / 40

Logic

Logic:Implications:

We then say that “P is equivalent to Q”.

because we have both “P if Q” and “P only if Q”, we also say “P if andonly if Q”.

We often write “P iff Q” for short.

The symbol ⇐⇒ is an equivalence arrow.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 34 / 40

Logic

Logic:Implications:

We then say that “P is equivalent to Q”.

because we have both “P if Q” and “P only if Q”, we also say “P if andonly if Q”.

We often write “P iff Q” for short.

The symbol ⇐⇒ is an equivalence arrow.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 34 / 40

Logic

Logic:Implications:

We then say that “P is equivalent to Q”.

because we have both “P if Q” and “P only if Q”, we also say “P if andonly if Q”.

We often write “P iff Q” for short.

The symbol ⇐⇒ is an equivalence arrow.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 34 / 40

Logic

Logic:Necessary and Sufficient Conditions:

There are other commonly used ways of expressing that proposition Pimplies proposition Q, or that P is equivalent to Q.

If proposition P implies proposition Q, we say that P is a “sufficient”condition for Q.

After all, for Q to be true, it is sufficient that P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 35 / 40

Logic

Logic:Necessary and Sufficient Conditions:

There are other commonly used ways of expressing that proposition Pimplies proposition Q, or that P is equivalent to Q.

If proposition P implies proposition Q, we say that P is a “sufficient”condition for Q.

After all, for Q to be true, it is sufficient that P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 35 / 40

Logic

Logic:Necessary and Sufficient Conditions:

There are other commonly used ways of expressing that proposition Pimplies proposition Q, or that P is equivalent to Q.

If proposition P implies proposition Q, we say that P is a “sufficient”condition for Q.

After all, for Q to be true, it is sufficient that P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 35 / 40

Logic

Logic:Necessary and Sufficient Conditions:

Accordingly, we know that if P is satisfied, then it is certain that Q is alsosatisfied.

In this case, we say that Q is a “necessary” condition for P.

Q must necessarily be true if P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 36 / 40

Logic

Logic:Necessary and Sufficient Conditions:

Accordingly, we know that if P is satisfied, then it is certain that Q is alsosatisfied.

In this case, we say that Q is a “necessary” condition for P.

Q must necessarily be true if P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 36 / 40

Logic

Logic:Necessary and Sufficient Conditions:

Accordingly, we know that if P is satisfied, then it is certain that Q is alsosatisfied.

In this case, we say that Q is a “necessary” condition for P.

Q must necessarily be true if P is true.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 36 / 40

Logic

Logic:Necessary and Sufficient Conditions:

This is summarized below:

P is a sufficient condition for Q means: P =⇒ QQ is a necessary condition for P means: P =⇒ Q

For example:

A necessary condition for x to be a rectangle is that x be a square.or

A sufficient condition for x to be a square is that x be a rectangle.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 37 / 40

Logic

Logic:Necessary and Sufficient Conditions:

This is summarized below:

P is a sufficient condition for Q means: P =⇒ QQ is a necessary condition for P means: P =⇒ Q

For example:

A necessary condition for x to be a rectangle is that x be a square.or

A sufficient condition for x to be a square is that x be a rectangle.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 37 / 40

Logic

Logic:Necessary and Sufficient Conditions:

This is summarized below:

P is a sufficient condition for Q means: P =⇒ QQ is a necessary condition for P means: P =⇒ Q

For example:

A necessary condition for x to be a rectangle is that x be a square.or

A sufficient condition for x to be a square is that x be a rectangle.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 37 / 40

Logic

Logic:Necessary and Sufficient Conditions:

This is summarized below:

P is a sufficient condition for Q means: P =⇒ QQ is a necessary condition for P means: P =⇒ Q

For example:

A necessary condition for x to be a rectangle is that x be a square.

orA sufficient condition for x to be a square is that x be a rectangle.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 37 / 40

Logic

Logic:Necessary and Sufficient Conditions:

This is summarized below:

P is a sufficient condition for Q means: P =⇒ QQ is a necessary condition for P means: P =⇒ Q

For example:

A necessary condition for x to be a rectangle is that x be a square.or

A sufficient condition for x to be a square is that x be a rectangle.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 37 / 40

Logic

Logic:Necessary and Sufficient Conditions:

The corresponding way to express P ⇐⇒ Q verbally is simply:

P is a necessary and sufficient condition for Q.

It is evident from this that it is very important to distinguish between thepropositions “P is a necessary condition for Q” and “P is a sufficientcondition for Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 38 / 40

Logic

Logic:Necessary and Sufficient Conditions:

The corresponding way to express P ⇐⇒ Q verbally is simply:

P is a necessary and sufficient condition for Q.

It is evident from this that it is very important to distinguish between thepropositions “P is a necessary condition for Q” and “P is a sufficientcondition for Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 38 / 40

Logic

Logic:Necessary and Sufficient Conditions:

The corresponding way to express P ⇐⇒ Q verbally is simply:

P is a necessary and sufficient condition for Q.

It is evident from this that it is very important to distinguish between thepropositions “P is a necessary condition for Q” and “P is a sufficientcondition for Q”.

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 38 / 40

Logic

Logic:Necessary and Sufficient Conditions:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 39 / 40

Logic

Logic:Necessary and Sufficient Conditions:

Alex Vickery (Royal Holloway) Math Boot Camp - Class #5 28th September, 2017 40 / 40