Relations and Functions

These flashcards cover essential definitions and concepts related to relations and functions, which are fundamental in mathematics.

Relation

A recognisable connection or link between two objects or quantities.

Domain

The set of all possible input values for a function.

Co-domain

The set of all possible output values for a function.

Range

The actual output values a function can produce.

Empty Relation

A relation R in a set A where no element of A is related to any element of A (R = φ).

Universal Relation

A relation R in a set A where each element of A is related to every element of A (R = A × A).

Reflexive Relation

A relation R in a set A where (a, a) ∈ R for every a ∈ A.

Symmetric Relation

A relation R in a set A where if (a₁, a₂) ∈ R, then (a₂, a₁) ∈ R for all a₁, a₂ ∈ A.

Transitive Relation

A relation R in a set A where if (a₁, a₂) ∈ R and (a₂, a₃) ∈ R, then (a₁, a₃) ∈ R for all a₁, a₂, a₃ ∈ A.

Equivalence Relation

A relation R in a set A that is reflexive, symmetric, and transitive.

One-one Function (Injective)

A function f: X → Y where distinct elements in X map to distinct elements in Y.

Onto Function (Surjective)

A function f: X → Y where every element of Y is the image of some element in X.

Bijective Function

A function that is both one-one and onto.

Composition of Functions

The combination of two functions f and g, denoted as g ∘ f, defined by (g∘f)(x) = g(f(x)).

Invertible Function

A function f that has an inverse function g such that f(g(y)) = y and g(f(x)) = x for all x in the domain and y in the co-domain.

Equivalence Class

The subset of X containing all elements that are related to a specific element under an equivalence relation.