Lecture 2.2: Set Theory

CS 250, Discrete Structures, Fall 2015

Nitesh Saxena

Adopted from previous lectures by Cinda Heeren

04/21/23 Lecture 2.2 -- Set Theory 2

Course Admin HW1

Was just due We will start to grade it We will provide a solution set soon

Word Equation editor; Open Office Travel next week

Attending and presenting at a conference in Vienna:

http://esorics2015.sba-research.org/ No class next week (Tuesday and Thursday)

Would not affect our coverage Please utilize this time to review the previous lectures

04/21/23 Lecture 2.2 -- Set Theory 3

Outline

Set Theory, Operations and Laws

04/21/23 Lecture 2.2 -- Set Theory 4

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

04/21/23 Lecture 2.2 -- Set Theory 5

Set Theory - Operators

A B = { x : (x A x B) v (x B x A)}

= (A - B) U (B - A)

Proof:

{ x : (x A x B) v (x B x A)}= { x : (x A - B) v (x B - A)}

= { x : x ((A - B) U (B - A))}

= (A - B) U (B - A)

04/21/23 Lecture 2.2 -- Set Theory 6

Set Theory - Famous Laws Two pages of (almost) obvious.

One page of HS algebra.

One page of new.

Don’t memorize

them, understand

them!

They’re in Rosen, p.

130

04/21/23 Lecture 2.2 -- Set Theory 7

Set Theory - Famous Laws Identity

Domination

Idempotent

A U = AA U = A

A U U = UA =

A U A = AA A = A

04/21/23 Lecture 2.2 -- Set Theory 8

Set Theory - Famous Laws Excluded Middle

Uniqueness

Double complement

A U A = U

A A =

A = A

04/21/23 Lecture 2.2 -- Set Theory 9

Set Theory – Famous Laws 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)

04/21/23 Lecture 2.2 -- Set Theory 10

Set Theory – Famous Laws DeMorgan’s I

DeMorgan’s II

p q

Venn Diagrams are good for intuition, but we aim for a

more formal proof.

(A U B) = A B

(A B) = A U B

04/21/23 Lecture 2.2 -- Set Theory 11

3 Ways to prove Laws or set equalities

Show that A B and that A B.

Use a membership table.

Use logical equivalences to prove equivalent set definitions.

New & important

Like truth tables

Not hard, a little tedious

04/21/23 Lecture 2.2 -- Set Theory 12

Example – the first wayProve that

1. () (x A U B) (x A U B) (x A and x B) (x A B)

2. () (x A B) (x A and x B) (x A U B) (x A U B)

(A U B) = A B

04/21/23 Lecture 2.2 -- Set Theory 13

Example – the second wayProve that using a

membership table.0 : x is not in the specified set1 : otherwise

(A U B) = A B

A B A B A B AUB A U B

1 1 0 0 0 1 0

1 0 0 1 0 1 0

0 1 1 0 0 1 0

0 0 1 1 1 0 1

04/21/23 Lecture 2.2 -- Set Theory 14

Example – the third wayProve that using

logically equivalent set definitions.(A U B) = A B

(A U B) = {x : (x A v x B)}

= {x : (x A) (x B)}

= A B

= {x : (x A) (x B)}

04/21/23 Lecture 2.2 -- Set Theory 15

Another example: applying the laws

X (Y - Z) = (X Y) - (X Z). True or False?

Prove your response.

= (X Y) (X’ U Z’)

= (X Y X’) U (X Y Z’)

= U (X Y Z’)

= (X Y Z’)

= X (Y - Z)

(X Y) - (X Z) = (X Y) (X Z)’

04/21/23 Lecture 2.2 -- Set Theory 16

Suppose to the contrary, that A B , and that x A B.

A Proof (direct and indirect)Pv that if (A - B) U (B - A) = (A U B) then

Then x cannot be in A-B and x cannot be in B-A.

But x is in A U B since (A B) (A U B).

A B =

Thus, A B = .

Then x is not in (A - B) U (B - A).

04/21/23 Lecture 2.2 -- Set Theory 17

Today’s Reading Rosen 2.1 and 2.2