Understanding Sets in Mathematics

• The concept of a set is fundamental in modern mathematics and is applied across various mathematical branches.
• Sets are crucial for defining relations and functions, and essential in studying geometry, sequences, and probability.
• Georg Cantor (1845-1918), a German mathematician, developed the theory of sets.
• Cantor initially encountered sets while working on trigonometric series problems.

Set Representation

• Sets are collections of objects; examples include a pack of cards, natural numbers, and prime numbers.
• Odd natural numbers less than 10: 1, 3, 5, 7, 9
• Vowels in the English alphabet: a, e, i, o, u
• Prime factors of 210: 2, 3, 5, and 7
• Solution of the equation x² – 5x + 6 = 0: 2 and 3
• A set must be a well-defined collection to ensure clarity about an object's inclusion.
• A well-defined set allows a definite determination of whether an object belongs to the collection.
• Examples of sets used in mathematics include N (natural numbers), Z (integers), Q (rational numbers), and R (real numbers).
• Z+ represents the set of positive integers.
• Q+ denotes the set of positive rational numbers.
• R+ is the set of positive real numbers.
• Terms like objects, elements, and members are synonymous when referring to sets.
• Sets are commonly denoted by capital letters like A, B, C, X, Y, Z.
• Elements of sets are usually represented by lowercase letters such as a, b, c, x, y, z.
• The symbol "∈" (epsilon) indicates that an element belongs to a set.
• The notation "a ∈ A" means 'a' belongs to set A.
• The symbol "∉" indicates that an element does not belong to a set
• The notation 'b ∉ A' means 'b' does not belong to set A.
• In the set V of vowels, a ∈ V but b ∉ V; in the set P of prime factors of 30, 3 ∈ P but 15 ∉ P.
• Sets can be represented in roster form or set-builder form.

Roster Form

• In roster form, elements are listed, separated by commas, and enclosed in braces { }.
• Example: The set of even positive integers less than 7 is {2, 4, 6}.
• The set of natural numbers dividing 42 is {1, 2, 3, 6, 7, 14, 21, 42}.
• The order of elements in roster form doesn't matter.
• Thus {1, 3, 7, 21, 2, 6, 14, 42} accurately represents the set of natural numbers that divide 42.
• In roster form, elements are not generally repeated; each element is taken as distinct.
• The set of letters forming the word 'SCHOOL' is {S, C, H, O, L}.
• The order of listing elements is irrelevant.

Set-Builder Form

• In set-builder form, all elements in a set share a common property, not found in any element outside the set.
• V = {x : x is a vowel in the English alphabet}
• In set-builder form, a symbol like x represents elements followed by a colon and a characteristic property within braces.
• A = {x : x is a natural number and 3 < x < 10} represents numbers between 3 and 10.

Examples

• The solution set of x² + x – 2 = 0 is {1, –2} in roster form.
• The set {x : x is a positive integer and x² < 40} in roster form is {1, 2, 3, 4, 5, 6}.
• A = {x : x is the square of a natural number} or A = {x : x = n², where n ∈ N} represents the set_A = {1, 4, 9, 16, 25, ... } in set-builder form.
• The set {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in set-builder form is {x : x = n/(n+1), where n is a natural number and 1 ≤ n ≤ 6}.
• Matching sets in roster and set-builder forms:
  • {P, R, I, N, C, A, L} matches {x : x is a letter of the word PRINCIPAL}.
  • {0} matches {x : x is an integer and x + 1 = 1}.
  • {1, 2, 3, 6, 9, 18} matches {x : x is a positive integer and is a divisor of 18}.
  • {3, –3} matches {x : x is an integer and x² – 9 = 0}.