Relations and Functions

This set of flashcards covers essential vocabulary and definitions related to relations and functions in mathematics.

Mathematical Beauty

The concept that, although hard to define, allows recognition of aesthetically pleasing mathematics.

Relation (in Mathematics)

A relation from set A to set B is defined as an arbitrary subset of A × B.

Empty Relation

A relation in a set A where no element of A is related to any element of A, denoted as R = φ ⊂ A × A.

Universal Relation

A relation in a set A where each element of A is related to every element of A, denoted as R = A × A.

Reflexive Relation

A relation R in a set A is reflexive if (a, a) ∈ R for every a ∈ A.

Symmetric Relation

A relation R in a set A is symmetric if (a1, a2) ∈ R implies (a2, a1) ∈ R for all a1, a2 ∈ A.

Transitive Relation

A relation R in a set A is transitive if (a1, a2) ∈ R and (a2, a3) ∈ R implies (a1, a3) ∈ R for all a1, a2, a3 ∈ A.

Equivalence Relation

A relation R in a set A is an equivalence relation if it is reflexive, symmetric, and transitive.

One-One Function (Injective)

A function f: X → Y is one-one (or injective) if distinct elements of X map to distinct elements of Y.

Onto Function (Surjective)

A function f: X → Y is onto (or surjective) if every element of Y is the image of some element in X.

Bijective Function

A function f: X → Y is bijective if it is both one-one (injective) and onto (surjective).

Composition of Functions

The composition of functions f: A → B and g: B → C is denoted as gof: A → C defined by gof(x) = g(f(x)) for all x ∈ A.

Function Inverses

A function f is invertible if there exists a function g such that gof = IX and fog = IY, with g denoted as f^(-1).

Equivalence Class

The equivalence class [a] containing a ∈ X for an equivalence relation R is the subset of X containing all elements related to a.

Trivial Relation

Refers to either the empty relation or the universal relation.

Examples of Relations

Provide examples like sibling relations, age comparisons, or locality to illustrate how relations can be defined.