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Math 3336 Section 2.1 Sets • • • • • • Definition of sets Describing Sets • Roster Method • Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets Definition: A set is an unordered collection of distinct objects. Examples: a. the students in this class b. the chairs in this room c. counting numbers Definition: The objects in a set are called the elements, or members of the set. A set is said to contain its elements. Notation: • The notation 𝑎𝑎 ∈ 𝐴𝐴 denotes that 𝑎𝑎 is an element of the set 𝐴𝐴. • If 𝑎𝑎 is not a member of 𝐴𝐴, write 𝑎𝑎 ∉ 𝐴𝐴. Ways to describe a set 1. Roster method • List all elements of a set • Order NOT important • Each distinct object is either a member or not; listing more than once does not change the set. • Ellipses (…) may be used to describe a set without listing all of the members when the pattern is clear. Example: 𝑆𝑆 = {𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑} Usually elements in a set have a common property but this is NOT necessary. Example: 𝐶𝐶 = {I, love, math, 10, Houston} 2. Set builder notation • Specify a property that all elements in a certain set must have. Example: 𝐵𝐵 = {𝑥𝑥 |𝑥𝑥 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 12 Page 1 of 4 Some important sets ℕ = natural numbers = {0,1,2,3 … . } ℝ = set of real numbers ℝ+ = set of positive real numbers ℤ = integers = {… , −3, −2, −1,0,1,2,3, … } ℤ+ = positive integers = {1,2,3, … . . } ℂ = set of complex numbers. ℚ = set of rational numbers Example: 𝐵𝐵 = {𝑥𝑥 |𝑥𝑥 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 12} Example: {ℕ, ℤ, ℚ, ℝ} Definition: Two sets are equal if and only if they have the same elements. Notation: 𝐴𝐴 = 𝐵𝐵 𝑖𝑖𝑖𝑖𝑖𝑖 ∀𝑥𝑥 (𝑥𝑥 ∈ 𝐴𝐴 ↔ 𝑥𝑥 ∈ 𝐵𝐵) Example: 𝐴𝐴 = {𝑟𝑟𝑟𝑟𝑟𝑟, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏}, 𝐵𝐵 = { 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏} Example: 𝐴𝐴 = {𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟, 𝑟𝑟𝑟𝑟𝑟𝑟}, 𝐵𝐵 = {𝑟𝑟𝑟𝑟𝑟𝑟} Venn Diagrams Sets can be represented graphically using Venn diagrams. Venn diagrams are often used to show relations between sets. John Venn (1834-1923) Cambridge, UK The universal set 𝑈𝑈 is the set containing everything currently under consideration. Sometimes implicit Sometimes explicitly stated. Contents depend on the context. The empty (null) set is the set with no elements. Symbolized ∅, but {} also used. Page 2 of 4 Example: Draw a Venn diagram that represents 𝑉𝑉 = {𝑎𝑎, 𝑒𝑒, 𝑖𝑖, 𝑜𝑜, 𝑢𝑢}. Example: 𝐴𝐴 = {𝑥𝑥 ∈ ℕ |𝑥𝑥 < 0} Definition: A set with one element is called a singleton set. Example: 𝐴𝐴 = {𝑥𝑥 ∈ ℕ|𝑥𝑥 ≤ 0} Definition: The set 𝐴𝐴 is a subset of 𝐵𝐵 if and only if every element of 𝐴𝐴 is also an element of 𝐵𝐵. 𝐴𝐴 is a strict (proper) subset of 𝐵𝐵 if 𝐴𝐴 is a subset of 𝐵𝐵 and 𝐴𝐴 is not equal to 𝐵𝐵. Notation: 𝐴𝐴 ⊆ 𝐵𝐵 and 𝐴𝐴 ⊂ 𝐵𝐵 respectively. Example: {1, 2} ⊂ {1, 2, 3, 4} ⊆ {1, 2, 3, 4} • • • To show that 𝐴𝐴 ⊆ 𝐵𝐵, show that if 𝑥𝑥 belongs to 𝐴𝐴, then x also belongs to 𝐵𝐵. To show that 𝐴𝐴 ⊈ 𝐵𝐵, find a single 𝑥𝑥 ∈ 𝐴𝐴 such that 𝑥𝑥 ∉ 𝐵𝐵. For every set 𝑆𝑆, ∅ ⊂ 𝑆𝑆 and 𝑆𝑆 ⊆ 𝑆𝑆. Prove: For every set 𝑆𝑆, ∅ ⊂ 𝑆𝑆. Proof: Definition: Given a set A, cardinality (size) of 𝐴𝐴 is the number of elements in 𝐴𝐴. Notation: |𝐴𝐴| Definition: The power set is the set of all subsets of a given set, and is denoted by 𝒫𝒫(𝐴𝐴). If a set has 𝑛𝑛 elements its power set contains 2𝑛𝑛 elements. Page 3 of 4 Try this: 𝐴𝐴 = {1, {2}, {3,4}}, 𝐵𝐵 = {1, 2, 3, 4} 1. True or False? a. 1 ∈ 𝐴𝐴 b. 2 ∈ 𝐴𝐴 c. {3} ⊂ 𝐴𝐴 d. 2 ∈ 𝐵𝐵 e. {3} ⊂ 𝐵𝐵 f. ∅ ∈ 𝐴𝐴 g. ∅ ⊂ 𝐵𝐵 h. 𝐴𝐴 ⊆ 𝐵𝐵 2. Find a. b. c. d. e. f. g. h. |𝐴𝐴| |𝐵𝐵| 𝒫𝒫(𝐴𝐴) 𝒫𝒫(𝐵𝐵) |∅| |{∅}| 𝒫𝒫(∅) 𝒫𝒫({∅}) Page 4 of 4