discrete mathematics. set theory - definitions and notation a set is an unordered collection of...
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Discrete Mathematics
Set Theory - Definitions and notation
A set is an unordered collection of elements.
Some examples:
{1, 2, 3} is the set containing “1” and “2” and
“3.”
{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is
irrelevant.
{1, 2, 3} = {3, 2, 1} since sets are unordered.
{1, 2, 3, …} is a way we denote an infinite set
= {} is the empty set, or the set containing no
elements.
Note: {}
Set Theory - Definitions and notation
x S means “x is an element of set S.”
x S means “x is not an element of set S.”
A B means “A is a subset of B.”
Venn Diagram
or, “B contains A.”or, “every element of A is also in B.”
or, x ((x A) (x B)).
A
B
Set Theory - Definitions and notation
A B means “A is a subset of B.”A B means “A is a superset of B.”
A = B if and only if A and B have exactly the same elements.
iff, A B and B Aiff, A B and A B iff, x ((x A) (x B)).
A B means “A is a proper subset of B.”
A B, and A B.
Set Theory - Definitions and notation
Quick examples:{1,2,3} {1,2,3,4,5}{1,2,3} {1,2,3,4,5}
Is {1,2,3}?Yes!
Is {1,2,3}?
No! Is
{,1,2,3}?Yes! Is
{,1,2,3}?Yes!
Set Theory - Definitions and notation
Quiz time:
Is {x} {x}?
Is {x} {x,{x}}?
Is {x} {x,{x}}?
Is {x} {x}?
Yes
Yes
Yes
No
Set Theory - Ways to define sets
Explicitly: {John, Paul, George, Ringo}
Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}
Set builder: { x : x is prime }, { x | x is odd }.
Set Theory - Cardinality
If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S.
If S = {1,2,3}, |S| = 3.If S = {3,3,3,3,3},
If S = ,
If S = { , {}, {,{}} },
|S| = 1.|S| =
0. |S| = 3.
If S = {0,1,2,3,…}, |S| is infinite.
Set Theory - Power sets
If S is a set, then the power set of S is
2S = { x : x S }.If S = {a},
aka P(S)
If S = {a,b},
If S = ,
If S = {,{}},
We say, “P(S) is the set of all subsets
of S.”
2S = {, {a}}.
2S = {, {a}, {b}, {a,b}}.
2S = {}.
2S = {, {}, {{}}, {,{}}}.
Set Theory - Cartesian Product
The Cartesian Product of two sets A and B is:A x B = { <a,b> : a A b B}
If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then
A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>}
Set Theory - Operators
The union of two sets A and B is:A B = { x : x A v x B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then
A B = {Charlie, Lucy, Linus, Desi}
AB
Set Theory - Operators
The intersection of two sets A and B is:
A B = { x : x A x B}If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then
A B = {Lucy}
AB
Set Theory - Operators
The complement of a set A is:A = { x : x A}
If A = {x : x is bored}, then
A = {x : x is not bored}
A= U and U =
U
Set Theory - Operators
The set difference, A - B, is:
AU
B
A - B = { x : x A x B }A - B = A B
Set Theory - Operators
The symmetric difference, A B, is:A B = { x : (x A x B) v (x
B x A)}= (A - B) U (B - A)
like “exclusive
or”
AU
B
Set Theory - Famous Identities
Identity
Domination
Idempotent
A U = AA U = A
A U U = UA = A
A U A = AA A = A
Set Theory - Famous Identities
Excluded Middle
Uniqueness
Double complement
A U A = U
A A =
A = A
Set Theory - Famous Identities
Commutativity
Associativity
Distributivity
A U B =
(A U B) U C =
A B =
B U A
B A
(A B) C =
A U (B U C)
A (B C)
A U (B C) = A (B U C) =
(A U B) (A U C) (A B) U (A C)
Set Theory - Famous Identities
DeMorgan’s I
DeMorgan’s II
(A U B) = A B
(A B) = A U B
Set Theory - Inclusion/Exclusion
Example:How many people are wearing a watch?How many people are wearing sneakers?
How many people are wearing a watch OR sneakers?
AB
|A B| = |A| + |B| - |A B|
Set Theory - Inclusion/Exclusion
Example:There are 217 cs majors.157 are taking cs125.145 are taking cs173.98 are taking both.
How many are taking neither?
217 - (157 + 145 - 98) = 13
125173
Set Theory - Generalized Inclusion/Exclusion
Suppose we have:
And I want to know |A U B U C|
A B
C
|A U B U C| = |A| + |B| + |C|
+ |A B C|
- |A B| - |A C| - |B C|