Relation And Function for Class 12

In real life, relation and function provide a link between two entities. We come across various patterns and links that describe relationships in our daily lives, such as the relationship between a parent and a son, brother and sister, and so on. We come across various number relationships in mathematics as well, such as x is less than y, line l is parallel to line m, and so on. Relations and functions connect elements of one set (domain) to those of another (codomain).

• Relations and Functions

Relations and functions define a mapping between two sets of data (inputs and outputs) that results in ordered pairs of the form. In algebra, the concepts of relationship and function are extremely significant. They are commonly utilized in both mathematics and everyday life. 

• Definition: Function and Relation

relations and functions can be defined as:

• Relations – The cartesian product A x B is a subset of a relation R from a non-empty set B. 
• Functions – A relation  between two sets A and B is said to be a function if every element in set A has exactly one image in set B. To put it another way, no two elements of B have the same pre-image.

• Relation and Function Representation:

The arrow representation, algebraic form, set-builder form, visually, roster form, and tabular form are all examples of how relations and functions can be expressed. 

Define the function f as A = {1, 2, 3} → B = {1, 4, 9} with f (1) = 1, f (2) = 4, and f (3) = 9. Let us now represent this function in several ways.

Set-builder form – {(x, y): f(x) = y2, x ∈ A, y ∈ B}

Roster form – {(1, 1), (2, 4), (3, 9)}

• Relation and Function-Related Terms

Now that we’ve learned what relation and function mean, let’s look at the definitions of a few terminologies linked to relations and functions that will help us better comprehend the concept:

• The cartesian product P Q is the set of all ordered pairs of elements from two non-empty sets P and Q, that is, P × Q = {(p, q) : p ∈ P, q ∈ Q}
• The domain of a relation R from a set A to a set B is the set of all initial elements of the ordered pairs in the relation R. It’s known as the input set or pre-images.
• The range of a relation R from a set A to a set B is defined as the set of all second elements of the ordered pairs in the relation R.
• The codomain of the relation R is defined as the entire set B in a relation R from a set A to a set B. Range ⊆ Codomain

• Relations and Function Types

There are various sorts of relations and functions, each with its own set of characteristics that distinguishes it from the others. Let’s look at the types of relations and functions listed below:

• Relations Types:

The following is a list of many sorts of relations:

• An empty relation is one that has no elements, meaning that no element of set A is mapped or linked to any element of A. R = ∅.
• A universal relation in a set A is one in which each member of A is connected to every other element of A, i.e., R = A x A. The whole relation is what it’s called.
• A relation R on A is said to be an identity relation if each member of A is connected to itself that is R = {(a, a): for all a belongs to A}.
• Inverse Relation – Define R as a relation between sets P and Q, i.e. R ∈ P × Q. If R-1 from set Q to set P is denoted byR-1 = {(q, p): (p, q) ∈ R}, it is said to be an inverse relation.
• Reflexive Relation – A binary relation R defined on a set A is said to be reflexive if for every a ∈ A, we have aRa, that is, (a, a) ∈ R.
• A binary relation R defined on a set A is said to be symmetric if and only if we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R for members a, b ∈  A.
• A relation R is transitive if and only if (a, b)  ∈ R and (b, c) ∈ R (a, c) R  ⇒ (a, c)  ∈ R for  a, b, c ∈ A
• Equivalence Relation – If and only if a relation R defined on a set A is reflexive, symmetric, and transitive, it is said to be an equivalence relation.

Functions types:

The following is a list of various types of functions:

• One-to-One Function – If each element of A is mapped to a distinct element of B, the function f: A→ B is said to be one-to-one. Injective Function is another name for it.
• Onto: If every element of B is the image of some element of A under f, a function f: A → B is said to be onto, i.e. there exists some element an in A such that f(a) = b for every b ∈ B. The function is onto if and only if the range of the function equals B.
• Many to One Function – The function f: A → B defines a many to one function in which more than one element of the set A is associated to the same element in the set B.
• Bijective Function – A bijective function is a function that is both one-to-one and an onto function.
• The constant function is defined as f(x) = K, where K is a real number. The same range value of K is produced for different values of the domain (x value) in constant function.
• Identity Function – An identity function is a function that returns the image of each element in a set B as the same element, e.g., g (b) = b ∀ b ∈ B. As a result, it has the form g(x) = x.

• Conclusion:

A relation between two sets is defined as a collection of ordered pairs containing one object from each set. If the ordered pair (x,y) is in the relation and the item x is from the first set and the object y is from the second set, the objects are said to be related. A type of relation is a function. However, it is permissible for an object x in the first set to be associated to more than one object in the second set in a relation.