# MATH135 Lecture Notes - Lecture 6: Disjoint Sets, If And Only If, Subset

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Published on 29 Sep 2015

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MATH 135 - Lecture 6: Subsets, Set Equality, Converse, and iff

Comparing Sets:

● Disjoint sets: S and T are said to be disjoint sets when S ∩ T = ϕ

○ Ex. S = {1, 2, 3}

T = {6, 8, 10}

Sets S and T are disjoint.

● Subsets: A set S is called a subset of a set T, and is written S T, when every element ⊆

of S belongs to T

○ Ex. 1 S = {4, 5}

T = {4, 5, 6}

S T⊆

○ How do we prove S T? → Prove the following implication: if ⊆x S, then ∈x T ∈

○ Ex. 2 Let A = {n : 4 | (∈ ℕ n - 3)} and B = {2k + 1 : k }. Prove A B.∈ ℤ ⊆

Solution: Let x A.∈

Thus, 4 | (x - 3)

x - 3 = 4k for some k ∈ ℤ

x = 4k + 3

x = 4k + 2 + 1

x = 2(2k + 1) + 1

Since k , therefore 2∈ ℤ k + 1 ∈ ℤ

Therefore x B.∈

○ A set S is called a superset of a set T, and written S T, if every element of T ⊇

belongs to S.

● Proper subsets: A set S is called a proper subset of a set T, and written S T, if every ⊊

element of S belongs to T and there exists at least one element in T which does not

belong to S.

○ Using the previous example, is A B? → Yes, there are values in B that are not ⊊

in A (ex. 5)

Set Equality:

● Saying that two sets S and T are equal, and writing S = T have exactly the same

elements.

● Ex. Two sets S and T are equal, written S = T where S and T have exactly the same

elements. How do we prove S = T?

○ Show that S T and T S⊆ ⊆

○x S ∈ ⇒ x T and ∈x T ∈ ⇒ x S∈

○ Let S = {-1, 1, 0} and T = {x |∈ ℝ x3 = x}

Show S T:⊆

Let x S. Then ∈x = -1, 1, 0.

13 = 1, (-1)3 = -1, 03 = 0

Therefore x3 = x and S T.⊆

## Document Summary

Math 135 - lecture 6: subsets, set equality, converse, and iff. Disjoint sets: s and t are said to be disjoint sets when s t = . Subsets: a set s is called a subset of a set t, and is written s. T, when every element of s belongs to t. Prove the following implication: if n - 3)} and b = {2k + 1 : k x. Thus, 4 | (x - 3) x - 3 = 4k for some k x = 4k + 3 x = 4k + 2 + 1 x = 2(2k + 1) + 1. A set s is called a superset of a set t, and written s. T, if every element of t belongs to s. Proper subsets: a set s is called a proper subset of a set t, and written s.