1.1 Introduction to Sets

Class: CSE 16 Subject: computer-science discrete-math Date: 2024-09-27 Teacher: Prof. Musacchio

Sets

Intro

• A set is a collection of things(elements)
  • E.g.: A = {1, 2, 3, 4} ⇒ RosterNotation
• If an element is in a set it’s denoted as 1 ∈ A(using prev example)
• Sets are considered equal if they contain the same elements
  • E.g.: A = {1, 2, 3, 4}, B = {2, 1, 3, 4} ∴ A = B ⇒ order doesn’t matter( basically a python set)
  • Duplicates also don’t affect equivalence
    • E.g.: A = {1, 2, 3, 4}, B = {1, 1, 3, 2, 4} ∴ A = B

Special Sets

• Natural Numbers = ℕ = {1,2,3,4,5,6,7,…}
• Integers = ℤ = {…,−3,−2,−1,0,1,2,3,4,…}
• Empty Set = {} or ∅
• Real Numbers = sqrt(2) ∈ $\mathbb{R}$
• Rational Numbers = Q = {i / n: n ∈ N, i ∈ Z}

Set properties

• Size or cardinality is the number of elements it has, and this number is denoted as |X|.
  • E.g.: X = {1, 2, ,3, 4} , |X| = 4
  • E.g.: |N| = ∞
  • E.g.: |∅| = 0

Question

• If n ∈ Z then n + 1 ∈ Z and n - 1 ∈ Z(False)
• If n ∈ Z then n + 1 ∈ Z(True)

Other Notation

Set builder

• set builder refers to a way to denote a series of elements that cannot be enclosed within {}
  • E.g.: E = {…,−6,−4,−2,0,2,4,6,…} (set of all real even numbers)
    • Set builder notation = E = {2n : n ∈ Z }
• Notation: { _ : condition(s)}
  • _ = prototype
  • : = read such that
• E.g.: {n: n ∈ N and n ⇐ 5 } ⇒ {1, 2, 3, 4, 5}

Rational Numbers

• E.g.: Q = {i / n: n ∈ N, i ∈ Z}