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Thursday, October 6, 2022

Study Types of Relations and Composite Functions Here

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Concepts like Relations and Function and their types are one of the important topics of set theory. Sets, Relations and Functions are all interconnected topics. Sets are collections of ordered elements, whereas relations and functions are actions on sets.

While a relation defines the connection between the two given sets, a function is a relationship between the set of inputs to the set of outputs. Also, several types of relations and functions express the links between the sets. 

On that note, let’s learn about various types of relations along with an important function type (Composite Functions) in detail for better understanding.

Relations 

In mathematics, a relation is a relationship between two or more sets of values.

Assume that x and y are ordered pair sets. And if set x is related to set y, the values of set x are referred to as domain, but the values of set y are referred to as range.

Example: For ordered pairs={(1,2),(-3,4),(5,6),(-7,8),(9,2)}

The domain is = {-7,-3,1,5,9}

And range is = {2,4,6,8}

Types of Relations

There are eight major types of relations, which are as follows:

Empty Relation

An empty connection (or void relation) is one in which no set items are related to one another. For example, if A = {1, 2, 3}, one of the void relations can be R = {x, y} where |x – y| = 8. For an empty relation,

R = φ ⊂ A × A

Universal Relation

A universal (or full) relation is one in which every element of a set is related to one another. For example, set A = {a, b, c}. R = {x, y} will now be one of the universal relations, where |x – y| = 0. For a universal relation,

R = A × A

Identity Relation

An identity relation relates every element of a set to itself. In a set A = a, b, c, for example, the identity relation will be I = {a, a}, {b, b}, {c, c}. For identity relation,

I = {(a, a), a ∈ A}

Inverse Relation

When a set has elements that are inverse pairings of another set, there is an inverse relation. If A = {(a, b), (c, d)}, then R-1 = {(b, a), (d, c)} is the inverse relationship. As a result, for an inverse relation,

R-1 = {(b, a): (a, b) ∈ R}

Reflexive Relation

Every element in a reflexive relation maps to itself. Consider the set A = {1, 2,} for example. A reflexive relation is written as R = {(1, 1), (2, 2), (1, 2), (2, 1)}. The reflexive relation is denoted by-

(a, a) ∈ R

Symmetric Relation

In a symmetric relation, if a=b is true, then b=a is also true. In other words, a relation R is symmetric if and only if (b, a)∈ R holds when (a,b) ∈R. 

For example, R = {(1, 2), (2, 1)} is a symmetric relation for a set A = {1, 2}. 

Transitive Relation

If (x, y) ∈R, (y, z) ∈R, then (x, z) ∈R is a transitive relation.

Equivalence Relation

An equivalence relation is reflexive, symmetric, and transitive simultaneously.

What is a Composite Function?

If we are given two functions, we can combine them to generate a new function. The procedures involved in performing this operation are comparable to those involved in solving any function for any given value. Composite functions are the name given to such functions.

A composite function is typically one that is written inside of another function. A function’s composition is accomplished by replacing one function with another.

For example, the composite function of f (x) and g (x) is f [g (x)] (x). “f of g of x” is how the composite function f [g (x)] is written. The inner function g (x) is referred to as an inner function, while the outer function f (x) is referred to as an outer function.

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