chapter 2: basic structures: sets, functions,jin/discrete10spring/l04.pdf · sets. informally: a ....
TRANSCRIPT
Sets
1
Sets
Informally: A set is a collection of (mathematical) objects,
with the collection treated as a single mathematical object.
•
real numbers, •
complex numbers, C
•
integers,
•
All students in our class
Examples:
Defining Sets
Sets can be defined directly:
e.g. {1,2,4,8,16,32,…},
{CSC1130,CSC2110,…}
Order, number of occurence
are not important.
e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B}
A set can be an element of another set.
{1,{2},{3,{4}}}
Defining Sets by Predicates
The set of prime numbers:
){ }(|x A P x
The set of elements, x, in A such that
P(x) is true.
Commonly Used Sets
•
N
= {0, 1, 2, 3, …}, the set of natural numbers•
Z
= {…, -2, -1, 0, 1, 2, …}, the set of integers•
Z+
= {1, 2, 3, …}, the set of positive integers•
Q
= {p/q
| p Z, q Z, and q ≠
0}, the set of rational numbers•
R, the set of real numbers
Special Sets
•
Empty Set (null set):
a set that has no elements, denoted by ф
or {}. •
Example: The set of all positive integers that are greater than their squares is an empty set.
•
Singleton set: a set with one element •
Compare: ф and
{ф}–
Ф: an empty set. Think of this as an empty folder–
{ф}: a set with one element. The element is an empty set. Think of this as an folder with an empty folder in it.
Venn Diagrams
•
Represent sets graphically•
The universal set U, which contains all the objects under consideration, is represented by a rectangle. The set varies depending on which objects are of interest.
•
Inside the rectangle, circles or other geometrical figures are used to represent sets.
•
Sometimes points are used to represent the particular elements of the set.
7
a
V eio
uU
{7, “Albert”, /2, T}
Membership
x A x is an element
of Ax is in
A
/2
{7, “Albert”,/2, T}
/3
{7, “Albert”,/2, T}
14/2
{7, “Albert”,/2, T}
7 2/3
Examples:
Containment
A B A is a subset of BA is contained in
B
Every element of A is also an element of B.
Examples:
R, {3}{5,7,3}
every set, A A
A is a proper subset of B
Set Equivalence
Two sets are equal if and only if they have the same elements. That is, if A and B are sets, then A and B are equal if and only
if
We write A = B if A and B are equal sets.
B). xA x(x
• Example:•
Are sets {1, 3, 5} and {3, 5,1} equal?
• Are sets {1, 3, 3, 3, 5, 5, 5, 5} and {1, 3, 5} equal?
Basic Operations on Sets
:: { | ( ) ( )}A B x x A x B union:
:: { | }A B x x A x B intersection:
Basic Operations on Sets
Basic Operations on Sets
difference: :: { | ( ) ( )}A B x x A x B
Basic Operations on Sets
:: { | } x D x A DA Acomplement:
Cardinality
•
Example:–
Let A be the set of odd positive integers less than 10. Then |A| = 5.
–
Let S be the set of letters in the English alphabet. Then |A| = 26.
–
Null set has no elements, | ф | = 0.
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Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.
Infinite Sets
•
Example: The set of positive integers is infinite.
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A set is said to be infinite if it is not finite.
Cardinality
•
Finding the cardinality of |A U B|:|A U B| = |A| + |B| -
|A ∩ B |•
Example: A = {1,3,5,7,9}, B = {5,7,9,11}|A U B| = |A| + |B| -
|A ∩ B |= 5 + 4 –
3 = 6
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Partitions of Sets
Two sets are disjoint
if their intersection is empty.
A collection of nonempty sets {A1
, A2
, …, An
} is a partition
of a set Aif and only if
A1
, A2
, …, An
are mutually disjoint.
Power Sets
( ) :: { |o }p w A S S A
pow( , ) , , , ,a b a b a b
power set:
Cartesian Products
•
Sets are unordered, a different structure is needed to represent
an ordered collections –
ordered n-tuples.
•
Two ordered n-tuples
are equal if and only if each corresponding pair of their elements is equal.–
(a1, a2,…, an
) = (b1, b2,…, bn
) if and only if ai
= bi
for i = 1, 2, …, n
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The ordered n-tuple (a1, a2,…, an
) is the ordered collection that has a1as its first element, a2
as its second element, …, and an
as its nth element.
Cartesian Products
•
Example: What is the Cartesian product of A = {1,2} and B = {a,b,c}?Solution:
A ×
B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}•
Cartesian product of A ×
B and B ×
A are not equal, unless A = ф or B
= ф (so that A ×
B = ф ) or
A = B. B ×
A = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}
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Let A and B be sets. The Cartesian product of A and B, denoted by A ×B, is the set of all ordered pairs (a, b), where a A and b B. Hence,
A ×
B = {(a,b)| a A Λ b B}.
Cartesian products
•
Example: What is the Cartesian product of A ×
B ×
C where A= {0,1}, B = {1,2}, and C = {0,1,2}?Solution:A ×
B ×
C= {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}
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The Cartesian product of sets A1, A2, …, An
, denoted by A1 × A2
× …×
An
is the set of ordered n-tuples
(a1, a2, …, an
), where ai
belongs to Ai
for i = 1,2, …, n. In other words,A1 × A2
× … × An = {(a1, a2, …, an
) | ai
Ai
for i = 1,2, …, n}.
Set Identities
Distributive Law:
Set Identities
De Morgan’s Law:
Proving Set Identities
Proving Set Identities
Computer Representation of Sets
•
Represent a subset A of U
with the bit string of length n, where the ith
bit in the string is 1 if ai
belongs to A and is 0 if ai
does not belong to A.
•
Example: –
Let U = {1,2,3,4,5,6,7,8,9,10}, and the ordering of elements of U has the elements in increasing order; that is ai
= i. What bit string represents the subset of all odd integers in
U?Solution: 10 1010 1010What bit string represents the subset of all even integers in U?Solution: 01 010 10101What bit string represents the subset of all integers not exceeding 5 in U?Solution: 11 1110 0000What bit string represents the complement of the set {1,3,5,7,9}?Solution: 01 0101 0101
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Logic and Bit Operations
•
Computers represent information using bits.•
A bit
is a symbol with two possible values, 0 and 1.•
By convention, 1 represents T (true) and 0 represents F (false).
•
A variable is called a Boolean variable if its value is either true or false.
•
Bit operation –
replace true by 1 and false by 0 in logical operations.
Table for the Bit Operators OR, AND, and XOR.x y x ν
y x Λ
y x y
0011
0101
0111
0001
0110
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Logic and Bit Operations
•
Example: Find the bitwise OR, bitwise AND, and bitwise XOR of the bit string 01 1011 0110 and 11 0001 1101.
DEFINITION 7A bit string is a sequence of zero or more bits. The length of this string is the number of bits in the string.
Solution:
01 1011 0110
11 0001 1101
-------------------
11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR
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Set Operations
•
The bit string for the union is the bitwise OR of the bit string for the two sets. The bit string for the intersection is the bitwise
AND
of the bit strings for the two sets.•
Example:–
The bit strings for the sets {1,2,3,4,5} and {1,3,5,7,9} are 11 1110 0000 and 10 1010 1010, respectively. Use bit strings to find the union and intersection of these sets. Solution:Union:
11 1110 0000 V 10 1010 1010
= 11 1110 1010, {1,2,3,4,5,7,9}Intersection:
11 1110 0000 Λ
10 1010 1010
= 10 1010 0000, {1,3,5}
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Let :: Sets |W S S S
S W S Sso
Russell’s Paradox
There is a male barber who shaves all those men, and only those men,who do not shave themselves.
Does the barber shave himself?