CSE20: Discrete Mathematics for Computer Science

Lecture Unit 8: Elementary Set Theory

Definition: Subset and proper subset

Definition: Set intersection and union

Set operations: Basic definitions

A �� B �� �x, such that x� A and x �� B

A � B �� (A � B) ���x� B such that x �� A

�

A � B =�

x : x� A and x� B

�

A � B =�

x : x� A or x� B

�

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 2

A � B �� �x, if x� A then x� B

A B

Definition: Set equalityA = B �� (A � B) � (B � A)

A B

A B

Set operations: Basic definitions

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 3

Definition: Set difference and symmetric differenceA� B = A \ B =

�x : x� A and x �� B

�

A� B =�

x : (x� A � x �� B) or (x� B � x �� A)�

A BA B

A� B = (A� B) � (B� A)Note:

A� B =�(x, y) : x� A and y� B

�Definition: Cartesian product of sets

A

k =�(x1, x2, . . . , x

k

) : x1, x2, . . . , x

k

� A

�

k

�i=1

A

i

=�(x1, x2, . . . , x

k

) : x1 � A1, x2 � A2, . . . , x

k

� A

k

�More generally:

More generally:

Algebraic rules for sets

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 4

P �Q = Q � P P �Q = Q � P

(P �Q) �R = P � (Q �R) (P �Q) �R = P � (Q �R)

(P �Q) � (P �R) = P � (Q �R)(P �Q) � (P �R) = P � (Q �R)(P �Q) � (P �R) = P � (Q �R)(P �Q) � (P �R) = P � (Q �R)

P � P = P P � P = P

P � (P �Q) = P P � (P �Q) = P

�P �Q

�c = P c �Qc�P �Q

�c = P c �Qc

Commutative Rule

Associative Rule

Distributive Rule

Idempotent Rule

Absorption Rule

DeMorgan Rule

In the table below P, Q, and R are arbitrary subsets of the same univer- sal set U; all the complements are with respect to U.

Venn diagrams

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 5

Venn diagrams for sets are similar to truth tables in propositional logic. They describe all possible intersections between several sets.

Q

R

P

PQ

P Q

PQcPcQ

PcQRPQcR

PQRc

PQR

PcQRcPQcRc

PcQcR

In general, a Venn diagram for n sets will have 2n distinct regions, similar to minterms in logic. For. example:.

PQ c

R

c =�

x � U : (x� P) � (x�Q c) � (x� R

c)�

Venn diagrams for 4 and 5 sets

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 6

Venn diagrams for sets are similar to truth tables in propositional logic. They describe all possible intersections between several sets.

A BC D

Beyond five sets, Venn diagrams become rather unwieldy.

BE

A

CD

Example: Prove that Q∩ (P∩R)c = Q∩ (P∩Q∩R)c Venn diagrams: Proving set identities

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 7

Q

R

P

1

2

3

4

8

6

7 5

(P∩Q∩R)c = { }1 2 3 4 5 6 7

P∩Q∩R = { }8

Q = { }2 5 6 8

(P∩R)c = { }1 2 3 4 5 6

2 5 6Q∩ (P∩Q∩R)c = { }Therefore:

7P∩R = { }8 Now:

Q = { }2 5 6 8

2 5 6Q∩ (P∩R)c = { } = Q∩ (P∩Q∩R)cTherefore:

Pascal Triangle Pascal triangle is a table and method to systematically compute the binomial coefficients for all n and k.(n

k)

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 1

�00

�=

�n0

�=

�nn

�= 1

Initialization:

�nk

�=

�n�1k�1

�+

�n�1

k

�Recursion:

k0 1 2 3 4 5 6 7 8

0 11 1 12 1 2 13 1 3 3 14 1 4 6 4 15 1 5 10 10 5 16 1 6 15 20 15 6 17 1 7 21 35 35 21 7 18 1 8 28 56 70 56 28 8 1

n

This triangle was described by Blaise Pascal in his Traité du Triangle Arith-métique (1653), though it was known to ancient Hindus, Greeks,and Chinese many centuries earlier (2-nd century BC).

Set partitionsLet A be an arbitrary set. A set partition of A is a set Ω ⊂ P(A) with the following properties:

Example: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

� =�{1}, {2}, {3, 5}, {4, 7}, {6, 8, 10}, {9, 11, 12}

�

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 9

X �Y = �

A =�

X��X

X �= �For all X∈Ω, we have1

For all X,Y∈Ω, either X =Y or2

The union of all X∈Ω is A itself: 3

The elements of a partition Ω ⊂ P(A) are called blocks. Note that blocks are themselves sets;. they are subsets of the original set A.

Example:

Counting partitions: Bell numbers The total number of different partitions of a set of size n is known as the Bell number B(n), defined as follows:

The Bell numbers B(n) and the Stirling numbers S(n,k) can be compu-ted recursively, in a manner similar to the Pascal triangle.

B(n) = S(n, 1) + S(n, 2) + · · · + S(n, n�1) + S(n, n)

Lecture Unit 8 CSE 20: Discrete Mathematics for Computer Science 10

k1 2 3 4 5 6 B(n)

1 1 12 1 1 23 1 3 1 54 1 7 6 1 155 1 15 25 10 1 526 1 31 90 65 15 1 203

n

S(n, 1) = S(n, n) = 1

Initialization:

S(n, k) =S(n�1, k�1) + kS(n�1, k)

Recursion: