Number Classification and Set Theory

Number Classification

• Natural numbers (ℕ) are positive integers starting from 1: {1, 2, 3, 4,...}
• Integers (ℤ) include positive and negative whole numbers and zero: {... , -2, -1, 0, 1, 2,...}
• Rational numbers (ℚ) are numbers that can be expressed as a fraction where numerator and denominator are integers: {𝑎/𝑏: 𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0}
• Irrational numbers (ℚ𝑐) are real numbers that are not rational.
• Real numbers (ℝ) consist of all rational and irrational numbers.

Set Notation and Representations

• {𝑥: 𝑥 is an integer and −3 < 𝑥 < 4} = {-2, -1, 0, 1, 2, 3}
• {𝑥: 𝑥 is a positive integer multiple of three} = {3, 6, 9, 12, ...}
• {𝑥: 𝑥 is an integer smaller than 5} = {... , -2, -1, 0, 1, 2, 3, 4}
• {𝑥: 𝑥 is a rational number with denominator 2} = {... , -3/2, -1/2, 0/2, 1/2, 3/2, ...}

Set Operations

• The Set *𝑥: 𝑥 = 𝑦^2 and 𝑦 is an integer = {0, 1, 4, 9, 16, ...}
• The set *𝑥: 𝑥 is an integer and (3𝑥 − 1)(𝑥 + 2) = 0 = {-2, 1/3}
• The set *𝑥: 2𝑥 is a positive integer = {1, 2, 3, 4, ...}
• If 𝑋 = {0, 1, 2}, then *𝑧: 𝑧 ∈ 𝑋 or −𝑧 ∈ 𝑋 = {0, 1, -1, 2, -2}
• The set *𝑥: 𝑥 is an integer and 1/8 < 𝑥 < 17/2 has the cardinality 8
• The set {3,6,9,12,15,...,27,30} = {𝑥: 𝑥 = 3𝑧 and 1 ≤ 𝑧 ≤ 10}
• The set {2,3,5,7,11,13,17,19,23,...} = {𝑥: 𝑥 is a prime number}

Set Relations

• 2 ∈ {1,2,3,4,5} - True
• {2} ∈ {1,2,3,4,5} - False
• 2 ⊆ {1, 2, 3, 4, 5} - True
• {2} ⊆ {1, 2, 3, 4, 5} - True
• ∅ ⊆ {∅, {∅}} - True
• {∅} ⊆ {∅, {∅}} - True
• 0 ∈ ∅ - False
• {1,2,3,4,5} = {5,4,3,2,1} - True

Subsets

• Subsets of {𝑎} = {∅, {𝑎}}
• Subsets of {𝑎, 𝑏} = {∅, {𝑎}, {𝑏}, {𝑎, 𝑏}}
• Subsets of {𝑎, 𝑏, 𝑐} = {∅, {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐}}
• A set with 𝑛 elements will have 2^𝑛 subsets
• The empty set has one subset, which is itself

Set Membership & Subset Relations

• 𝑥 = {1}; 𝐴 = {1, 2, 3} - 𝑥 ∈ 𝐴
• 𝑥 = {1}; 𝐴 = {{1}, {2}, {3}} - 𝑥 ⊆ 𝐴
• 𝑥 = {1}; 𝐴 = {1, 2, {1, 2}} - 𝑥 ∈ 𝐴 and 𝑥 ⊆ 𝐴
• 𝑥 = {1, 2}; 𝐴 = {1, 2, {1, 2}} - 𝑥 ∈ 𝐴 and 𝑥 ⊆ 𝐴
• 𝑥 = {1}; 𝐴 = {{1, 2, 3}} - Neither 𝑥 ∈ 𝐴 nor 𝑥 ⊆ 𝐴
• 𝑥 = 1; 𝐴 = {{1}, {2}, {3}} - Neither 𝑥 ∈ 𝐴 nor 𝑥 ⊆ 𝐴

Set Operations - Union, Intersection, Complement, Difference

• Let ℧ = {1,2,3,4,5,6,7,8,9,10}, 𝐴 = {𝑥 ∈ ℧: 𝑥 is less than 7}, 𝐵 = {𝑥 ∈ ℧: 𝑥 is a multiple of 3}

• 𝐴 ∪ 𝐵 = {1,2,3,4,5,6,9}

• 𝐴 ∩ 𝐵 = {3, 6}

• 𝐴𝑐 = {7, 8, 9, 10}

• 𝐵𝑐 = {1, 2, 4, 5, 7, 8, 10}

• 𝐴\B = {1, 2, 4, 5}

Set Relationships

• Let ℧ = {𝑥: 𝑥 is an integer and 2 ≤ 𝑥 ≤ 10}, 𝐴 = {𝑥: 𝑥 is odd}, 𝐵 = {𝑥: 𝑥 is a multiple of 3}

• 𝐴 ⊂ 𝐵 is not true

• 𝐵 ⊂ 𝐴 is not true

• 𝐴 = 𝐵 is not true

• Neither 𝐴 ⊂ 𝐵, 𝐵 ⊂ 𝐴 nor 𝐴 = 𝐵 is true

• Let ℧ = {𝑥: 𝑥 is an integer and 2 ≤ 𝑥 ≤ 10}, 𝐴 = {𝑥: 𝑥 ∈ ℤ}, 𝐵 = {𝑥: 𝑥 is a power of 2 or 3}

• 𝐴 ⊂ 𝐵 is not true

• 𝐵 ⊂ 𝐴 is not true

• 𝐴 = 𝐵 is not true

• Neither 𝐴 ⊂ 𝐵, 𝐵 ⊂ 𝐴 nor 𝐴 = 𝐵 is true

Student Survey Problem

• There are 18 students in a room
• 7 study mathematics, 10 study science, and 10 study computer programming.
• 3 study mathematics and science, 4 study mathematics and computer programming, and 5 study science and computer programming.
• 1 student studies all three subjects.
• It can be concluded that 4 students study none of the three subjects.