Relations and Functions: Understand types and Difference

"Relations and Functions" stand out as crucial topics within algebra. These terms, though related, hold distinct mathematical meanings. It's common to feel uncertain about their disparities. To clarify, let's begin with a straightforward example.

An ordered pair is typically expressed as (INPUT, OUTPUT):

A relation indicates the connection between INPUT and OUTPUT. Conversely, a function is a relation where each given INPUT corresponds to precisely one OUTPUT.

Note: All functions are relations, but not all relations are functions. 

What is a Function?

A function is a specific type of relation where each input is associated with only one output. In other words, it adheres to a rule whereby every X-value corresponds to a single y-value. 

Types of Functions 

In terms of relations, functions can be categorized as follows:

1. One-to-One Function or Injective Function: A function f: P → Q is termed one-to-one if each element of P corresponds to a distinct element of Q.
2. Many-to-One Function: A function maps two or more elements of P to the same element in set Q.
3. Onto Function or Surjective Function: A function is onto if every element in set Q has a pre-image in set P.
4. One-to-One Correspondence or Bijective Function: This function, denoted as f, pairs each element of P with a unique element of Q, and every element of Q has a pre-image in P.

What is the Relation?

It's a subset of the Cartesian product or, put simply, a collection of points represented by ordered pairs. In essence, the relationship between two sets is established through this collection of ordered pairs, where each pair consists of one object from each set. For instance, {(-2, 1), (4, 3), (7, -3)}, typically denoted in set notation with curly brackets.

Relation Representation

Besides set notation, relations can also be represented in various other formats, including tables, plotting on the XY-axis, or using a mapping diagram.

Types of Relations

Different types of relations are as follows:

• Empty Relations
• Universal Relations
• Identity Relations
• Inverse Relations
• Reflexive Relations
• Symmetric Relations
• Transitive Relations

Empty Relation

When no element of set X is related or mapped to any other element within X, the relation R in A becomes an empty relation, also known as the void relation, denoted as R = ∅.

Universal relation

Let's consider R as a relation within a set, where A represents the universal relation. In this complete relation, denoted as R = A × A, every element of set A is related to every other element of A. This comprehensive relation is termed a full relation since every element of set A is also a part of set B.

Identity Relation

When every element of set A is related only to itself, it is referred to as the Identity relation.

For instance,

Consider the outcomes when rolling a dice, totaling 36 possibilities such as (1, 1), (1, 2), (1, 3), ..., (6, 6). If we isolate the relation (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6), it constitutes an identity relation.

Inverse Relation

If R is a relation from set A to set B, denoted as R ∈ A × B, then the relation R^-1 is defined as {(b, a) : (a, b) ∈ R}.

For instance,

Consider the scenario of throwing two dice. If R = {(1, 2), (2, 3)}, then R^-1 = {(2, 1), (3, 2)}. Here, the domain of R is equivalent to the range of R^-1, and vice versa.

Reflexive Relation

A relation is a reflexive relation if every element of set A maps to itself, i.e for every a ∈ A, (a, a) ∈ R.

Symmetric Relation

A symmetric relation is a relation R on a set A if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation

If (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a,b,c ∈ A and this relation in set A is transitive.

Equivalence Relation

If a relation is reflexive, symmetric and transitive, then the relation is called an equivalence relation.