6.1 Set Theory: Definitions and the Element Method of Proof

Notation

• A = {x ∈ S | P(x)} read as "A is the set of all x in S such that P of x".
• Venn Diagrams
• Universal Set: U
• Empty Set: ∅
• Interval Notation: e.g. (a, b] = {x ∈ R | a < x ≤ b}

Definitions

• Set A is a subset of Set B: A ⊆ B ⇔ ∀ x, if ∈ A then x ∈ B.
• Set A is a proper subset of Set B: A ⊆ B and ∃ x, x ∈ B and x ∉ A.
• Set Equality: A = B ⇔ A ⊆ B and B ⊆ A.
• Union: A ∪ B = {x ∈ U | x ∈ A or x ∈ B}
• Intersection: A ∩ B = {x ∈ U | x ∈ A and x ∈ B}
• Set Difference: A - B = {x ∈ U | x ∈ A and x ∉ B}
• Complement: A^c = {x ∈ U | x ∉ A}
• Disjoint: A and B are disjoint ⇔ A ∩ B = ∅
• Mutually Disjoint Sets: A_i ∩ A_j = ∅ whenever i ≠ j
• Partition of A: A is the union of all A_i and the A_i sets are mutually disjoint
• Power Set of A: P(A) is the set of all subsets of A

Element Argument: Used to Prove X ⊆ Y

1. Suppose that x is a particular, but arbirtrarily chosen element of X.
2. Show that x is an element of Y.

In Class Exercises

13

• Let S be the set of all strings of 0's and 1's of length 4
• Let A = {1110, 1111, 1000, 1001}
• Let B = {1100, 0100, 1111, 0111}
• Find A ∪ B, A ∩ B, A - B, A^c

15

• Draw a Venn Diagram that satisfies (1) A ∩ B = ∅, (2) A ⊆ C and (3) C ∩ B ≠ ∅

29

• Is {R⁺, R^-, 0} a partition of R? Explain.

31

• Let A = {1,2} and B = {2,3}
• Find P(A ∩ B)
• Find P(A)

6

• Let B = {y ∈ Z | y = 10b - 3 for some integer b}
• Let C = {z ∈ Z | z = 10c + 7 for some integer c}
• Prove B = C