Relations in Mathematics

What is a Relation?

• A relation exists when two sets are defined with a function between them.
• The Cartesian product of the sets determines if the relation is valid.
• Three key concepts in relations are reflexivity, symmetry, and transitivity.

Types of Relations

• A relation is reflexive if, for every element x in set A, the pair (x, x) is in the relation.
• A relation is symmetric if, for every (a, b) in the relation, (b, a) must also be in the relation.
• A relation is transitive if, for every (a, b) and (b, c) in the relation, (a, c) must also be in the relation.
• An equivalence relation satisfies all three properties (reflexive, symmetric, transitive).

Domain, Range, and Codomain

• The domain of a relation is the set of all "first" elements of the pairs in the relation.
• The range of a relation is the set of all "second" elements of the pairs in the relation.
• The codomain of a relation is the set of all possible "second" elements; usually the right-hand set in the relation.

Examples & Properties of Relations

• Example 1: Set A = {1, 2, 3, 4, 5, 6, 7}, Set B = {1, 2, 3, 4, 5, 6, 7, 8, 9,...}. Function: a = x².

  • Relation R: (1, 1), (2, 4), (3, 9), ...
  • Domain of R: {1, 2, 3, 4, 5, 6, 7}
  • Range of R: {1, 4, 9, 16, 25, 36, ...}
  • Codomain of R: All natural numbers (including 0).
  • R is not reflexive, symmetric, or transitive.

• Example 2: Set A = {1, 2}, Set B = {1, 2, 3, 4}. Relation R = {(1, 1), (1, 2), (2, 1), (2, 3), (2, 4)}.

  • R is reflexive (1, 1) and (2, 2) are present but not for all elements
  • R is symmetric (1, 2) and (2, 1) are in R.
  • R is not transitive (1, 2) and (2, 3) are in R, but (1, 3) is not.

• Example 3: Set A = {1, 2, 3, 4}, Relation R based on a = x².

  • R is not reflexive, symmetric, or transitive.

• Example 4: Set A = {1, 2, 3, 4, 5,...}. Relation R: (a, b) where (a - b) is divisible by 3.

  • R is reflexive (a - a = 0 is divisible by 3).
  • R is symmetric (if a - b is divisible by 3 then b - a is also divisible by 3).
  • R is transitive (if a - b and b - c are divisible by 3 then a - c is divisible by 3).

Key Ideas

• Divisibility is crucial in determining transitive relations.

Applying the Concepts

• To identify an equivalence relation:
  1. Check for reflexivity.
  2. Check for symmetry.
  3. Check for transitivity.
  • If all three conditions are met, it's an equivalence relation.