Relations and Functions

Study Notes

There's no universally accepted definition of mathematical beauty, but that applies to all types of beauty.
Relations and functions were introduced in Class XI.
Relations are drawn from the English language concept of connections between two objects.
A relation in set A to set B refers to any subset of the Cartesian product A x B.
A relation from A to B, denoted by R, is a subset of A × B. If (a, b) ∈ R, then a is related to b under the relation R.
Empty relation and Universal relation are some times trivial.
Reflexive relation: (a, a) ∈ R for all a ∈ A
Symmetric relation: If (a₁, a₂) ∈ R, then (a₂, a₁) ∈ R
Transitive relation: If (a₁, a₂) ∈ R and (a₂, a₃) ∈ R, then (a₁, a₃) ∈ R
An equivalence relation is reflexive, symmetric, and transitive.
Relations can be represented using roster method and set builder method.
A relation in a set A is said to be an equivalence relation if it is reflexive, symmetric, and transitive.
Example illustrating relations are given, including the set of students in classes XII and XI of a school.
A relation R in set A is said to be reflexive if (a, a) ∈ R for all a ∈ A.
A relation R in a set A is said to be symmetric if (a,b) ∈ R implies (b, a) ∈ R.
A relation R in a set A is said to be transitive if (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R
Examples are given showing that some relations are reflexive, symmetric, or transitive.

Empty Relation: No element of A is related to any element of A (R = ∅).
Universal Relation: Every element of A is related to every element of A (R = A × A).
A relation R in set A is called empty relation, if no element of A is related to any element of A, i.e., R = ¢ ⊂ A × A.
A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.
Example showing that a relation in a set of students in a boys school can be an empty relation.

A function f : X → Y is defined as a relation such that each element in X is mapped to exactly one element in Y.
A function f: X → Y is said to be one-one (or injective) if the images of distinct elements of X under f are distinct: For every x₁, x₂ ∈ X, if f(x₁) = f(x₂), then x₁ = x₂.
A function f : X → Y is said to be onto (or surjective) if each element y ∈ Y is the image of some element x in X under f: For every y ∈ Y, there exists x ∈ X such that f(x) = y.
A function f: X → Y is said to be one-one and onto (or bijective) if f is one-one and onto.
A function f : X → Y is called injective or one-one or both if and only if for every x₁, x₂ ∈ X, if f(x₁) = f(x₂), then x₁ = x₂.
A function f: X → Y is called surjective or onto or both if and only if the range of f equals the codomain Y.