Understanding Sets and Set Notations

• "Achieving Universal Understanding and Peace Through the Language of Mathematics"

Module Objectives

• Discussing the definition of a set in mathematics
• Identifying set symbols and notations
• Writing sets using two methods
• Differentiating various kinds of sets
• Showing the union and intersection of sets
• Set theory is a fundamental building block in mathematics, underpinning higher disciplines such as Graph Theory, Abstract Algebra, and Number Theory

Set and Set Notations

• Sets in mathematics are similar to everyday collections or groups
• A set is a collection of distinct objects, and each object must be well-defined
• Well-defined means it's clear whether an object belongs to the set, and distinct means no duplicates
  • Well-defined sets: the set of female presidents of the Philippines, the set of quadrilaterals
  • Not well-defined sets: the set of good Filipino writers, the set of best books in the library

Rules For Writing Sets

• Sets denoted by capital letters (A, B, C, ..., X, Y, Z)
• Elements are objects within a set, written in lowercase letters (a, b, c, ..., x, y, z)
• Elements are enclosed in braces { }
  • For example, Set A containing letters from "freshmen" is A = {f, r, e, s, h, m, n}

Set Membership

• '∈' symbol indicates an element is part of a set.
  • If f is an element of set A: f ∈ A.
• '∉' symbol indicates an element is not part of a set.
  • If a is not an element of set A: a ∉ A.

Set of Real Numbers

• Natural Numbers (N): Counting numbers, positive integers (1, 2, 3, ...)
• Integers (Z): Natural numbers with their negatives and 0 (...-4, -3, -2, -1, 0, 1, 2, 3, 4...)
• Rational Numbers (Q): Numbers expressible as a/b, where a and b are integers and b ≠ 0; decimal representations are either terminating or repeating (-15, -2, 0, -1/4, 3/7, -2.75, 1.625, -0.3333, 5.272727)
• Irrational Numbers (Q'): Non-repeating, non-terminating decimals (√2= 1.414213562..., π =3.141592654...)
• Real Numbers (R): Rational and Irrational numbers

Methods of Writing a Set

• Enumeration/Roster Method
  • Listing elements in any order without repetition, enclosed in curly braces
• Defining/Rule Method
  • Defining members by stating a common property
  • Set-builder notation: {x | P(x)} or {x: P(x)}, meaning "the set of all x such that"
  • Examples:
    • B = {11, 13, 17, 19, 23, 29} defined as B = {x | x is a prime number between 10 and 30}

Kinds of Sets

• Finite Set
  • Elements are limited/countable
  • Cardinality: the number of elements in a set, denoted by n(A)
    • A = {x | x is a positive integer less than 17} = {1, 2, 3, ..., 16}, n(A) = 16
• Infinite Set
  • Elements are unlimited/cannot be counted
• Unit/Singleton Set
  • Containing only one element
• Empty/Null Set
  • Containing no object/element with symbols "{ }" and "Ø"
• Equal Sets
  • Two sets with the same elements
  • Represented symbolically as A = B
• Equivalent Sets
  • Sets with the same number of elements/cardinality
  • Symbol for set equivalence is ≈

Disjoint Sets

• Two sets with no common elements

Subsets and Power Sets

• A is a subset of B (A ⊆ B) if every element of A is in B
• Ways to express that A is a subset of B are such as:
  • A is contained in B
  • B contains A
• Number of subsets in a set = 2^n
• Supersets
  • If A is a subset of B, then B is a superset of A, denoted by B ⊇ A
• Power Sets
  • Set of all subsets of A denoted as P(A)
  • Cardinality of the power set is |P(A)| = 2^n
• Universal Set
  • A set containing all possible elements for consideration
• Complementary Sets
  • Sets A and B are complementary if they share no common elements and their union is the universal set
    • A^c denotes a complement of set A

Operations of Sets

• Union
  • The union of sets A and B (A U B) includes all elements from both sets
• Intersection
  • The intersection of sets A and B (A ∩ B) includes only the elements common to both sets
• Product Sets
  • The product set of non-empty sets A and B is the set of all ordered pairs (a, b) where ‘a’ is from A and 'b’ is from B
  • Denoted using the symbol "x" and is read “A cross B.”
  • A x B ≠ B x A