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A relation is a collection of ordered pairs, which contains an object from one set to the other set. For instance, X and Y are two sets, and ‘a’ is the object from set X and b is the object from set Y, then we can say that the objects are related to each other if the order pairs of (a, b) are in relation.

Consider two arbitrary sets X and Y. The set of all ordered pairs (x,y) where x∈X and y∈Y is called the Cartesian product, of X and Y. The product is designated as, read as X cross Y. By deﬁnition,

X х Y = {(x, y)} I x ∈ X and y ∈ Y}

X х Y ≠ Y х X . The Cartesian product deals with ordered pairs, so the order in which the sets are considered is important. Using n(A) for the number of elements in a set A, we have:

n(X х Y) = n(X) х n(Y)

A relation from a set X to a set Y is any subset of the Cartesian product X×Y

A relation X to Y is a subset of X Y.

Let X and Y set. An ordered pair(x,y) is called a relation in x and y. The first element in an ordered pair is called the domain, and the set of second elements is called the range of the relation.

Let us consider R as a relation from X to Y. Then R is a set of ordered pairs where each ﬁrst element is taken from X and each second element is taken from Y. That is, for each x ∈ X and y ∈ Y, follows exactly one of the following:

i. x, y R; then“x is R-related to y”, written as xRy.

ii. x, y ∉ R; then “x is not R-related to y”, written as xRy

If R is a relation from a set X to itself, that is, if R is a subset of X2 =X X, we say that R is a relation on X.

The relation that defines the set of input elements to the set of output elements is called a function. Each input element in the set X has exactly one output element in the set Y in a function. A function requires two conditions to be satisfied to qualify as a function:

1. Every x∈X must be related to y∈Y, i.e., the domain of f must be X and not a subset of X.

2. There is a requirement of uniqueness, which can be expressed as:

(x,y) ∈f and (x,z) ∈f ⇒ y = z

Sometimes we represent the function with a diagram: f : A⟶B or Af⟶B

Functions are sometimes also called mappings or transformations.

To understand the difference between a relationship that is a function and a relation that is not a function. All functions are relations, but not all relations are functions. The difference between a relation and a function is that a relationship can have many outputs for a single input, but a function has a single input for a single output. This is the basic factor to differentiate between relation and function. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

A Relation is a group of ordered pairs of elements. It can be a subset of the Cartesian product. It is a dyadic relation or a two-place relation. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

Function pertains to an ordered triple set consisting of X, Y, F. X, where X is the domain, Y is the co-domain, and F is the set of ordered pairs in both “a” and “b.” Each ordered pair contains a primary element from the “A” set. The second element comes from the co-domain, and it goes along with the necessary condition.

In a set B, it pertains to the image of the function. The domain and co-domain are both sets of real numbers. It doesn’t have to be the entire co-domain. It can be known as the range. Relations show the properties of items. In a way, some things can be linked in some way, so that’s why it’s called “relation.” It doesn’t imply that there are no in-betweens that can distinguish between relation and function.

One thing good about it is the binary relation. It has all three sets. It includes the X, Y, and G. X and Y are arbitrary classes, and the “G” would have to be the subset of the Cartesian product, X x Y. They are known as the domain set of departure or even co-domain. G would be understood as a graph.

The function can be an item that takes a mixture of two-argument values that can give a single outcome. There is another difference between relation and function. The function should have a domain that results from the Cartesian product of two or more sets but is not necessary for relations.

a. Log functions can be written as exponential functions.

b. Logs of products involve addition and products of exponentials involve addition.

FAQ (Frequently Asked Questions)

Q.1 What is the Basic Difference Between Relation and Function in Math?

Ans: A relation represents the relationship between the input and output elements of two sets whereas a function represents just one output for each input of two given sets.

Q.2 What is Domain and Range in Function?

Ans: Domain is a set of all input elements of a set and range is a set of all output elements of a set.

Q.3 What are the Different Types of Functions in Maths?

Ans:

1. One-to-One or Injective

2. Onto or Surjective

3. One-To-One Correspondence or Bijective

4. Inverse functions

5. Logarithmic and exponential functions are two special types of functions.

6. Linear function

7. Quadratic function

8. Polynomial function