Discrete Mathematics - Lecture 2.2 Set Operations PDF

Preview

Summary

Discrete Mathematics - Lecture 2.2 Set Operations...

Description

Math 3336 Section 2.2 Set Operations •

Set Operations • Union • Intersection • Difference • Complement

• • • •

More on Set Cardinality Set Identities Proving Identities Membership Tables

Definition: Let ฀฀ and ฀฀ be sets contains those elements that are

, is the set that

Try this one: Write the union of ฀฀ and ฀฀ in the set builder notation.

U A

B Venn diagram for ฀฀ ∪ ฀฀

Definition: Let ฀฀ a containing those ele Two sets are called

of the sets ฀฀ and ฀฀ , denoted, empty.

Try this one: Write the intersection of ฀฀ and ฀฀ in the set builder notation.

Page 1 of 6

, is the set

U A

B Venn diagram for ฀฀ ∩ ฀฀

Definition: Let ฀฀ and ฀฀ be sets. The containing the complement of ฀฀ with respect to ฀฀.

of ฀฀ and ฀฀ , denoted by , is the set The difference of ฀฀ and ฀฀ is also called the

Try this one: Write the difference of ฀฀ and ฀฀ in the set builder notation.

Definition: Let ฀฀ be the universal set. The c f the set A, denoted by is the complement of ฀฀ with respect to ฀฀. Therefore, the complement of the set ฀฀ is ฀฀ − ฀฀. Try this one: Write ฀฀, in the set builder notation.

umbers and the universal set is the set of natural

Page 2 of 6

U A Venn diagram for ฀฀ The cardinality of the union of two sets nclusion-exclusion principle Example: Find |฀฀ ∪ ฀฀| if ฀ ฀ = {1,2,3} and ฀ ฀ = {2, 3, 4, }.

Definition: Let A and B be sets. The set (฀฀ − ฀฀ ) ∪ (฀฀ − ฀฀ ).

of ฀฀ and ฀฀, denoted by

U A

B Venn diagram for symmetric difference

Try this one: ฀ ฀ = {฀฀ ∈ ℕ | ฀฀ ≤ 10}, ฀ ฀ = {1,2, }, and ฀ ฀ = { , 4, 5}. Find ฀฀⨁฀฀ .

Page 3 of 6

is the

Set Identities •

Identity laws:

•

Domination laws:

•

Idempotent laws:

•

Complementation law:

•

Commutative laws:

•

Associative laws:

•

Distributive laws:

•

De Morgan’s laws:

•

Absorption laws:

•

Complement laws:

Proving set identities Different ways to prove set identities • identity) is a •

t elements in same side of .

Page 4 of 6

on of sets always either to indicate it is

Example: Construct a membership table to show

Page 5 of 6

Example: Prove subsets of each other.

by showing that RHS set and LHS set are

Page 6 of 6...