Relations and Functions Quiz

Created by

@LuckierIndigo

Study Notes

Relations and Functions

A relation is a subset of the Cartesian product of two sets. The Cartesian product of sets A and B is written as A × B.

Types of Relations

Reflexive Relation: A relation R on a set A is reflexive if (a, a) ∈ R for every element a in A.
Identity Relation: A relation R on a set A is an identity relation if (a, a) ∈ R for every a in A, and no other elements are related. Basically, it only relates each element to itself.
Symmetric Relation: If (a, b) ∈ R, then (b, a) ∈ R. This means the relation holds both ways.
Transitive Relation: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. This means if a relates to b, and b relates to c, then a relates to c.

Number of Relations

If a set A has n elements, there are 2^n² possible relations on A.

Functions

A function is a specific type of relation between a domain (inputs) and a range (possible outputs).

Types of Functions

Injective Function (One-to-one): Every element in the range has at most one pre-image in the domain. No two different inputs map to the same output.
Surjective Function (Onto): Every element in the range has at least one pre-image in the domain. Every element in the codomain is hit by the function.
Bijective Function (One-to-one and Onto): A function that is both injective and surjective. Each output is associated with exactly one input, and every element in the range is used.

Number of Functions

If set A has n elements and set B has m elements, the number of functions from A to B is mⁿ. This formula only holds for finite sets.

Equivalence Relations

An equivalence relation combines the properties of reflexive, symmetric, and transitive relations.

Description

Test your understanding of relations and functions with this quiz. Dive into topics such as types of relations, properties of functions, and the Cartesian product. Perfect for students looking to solidify their grasp on these foundational concepts in mathematics.