Relations and Functions Class 11 Basic Concepts and Types of Relations and Functions

It is importat to understand Relations in Maths since this topic is covered in both CBSE board classes 11th and 12th. We will understand this through an example. Let us consider that A and B are any two non-empty sets. Then any subset R of A  B is called a relation from set A to set B. Thus, R is a relation from A to B if and only if R $\subseteq A\times B.$  

If R is a relation from A to B if $(a,b)\in R$ , then we write aRb and say that a is related to b and if $(a,b)\notin R$ , then write $(a,b)\notin R$ and say that a is not related to b. Students preparing for IISER and CUET must understand this concept as well since questions are asked around this chapter.

Example of Relations

Let  $N$ be the set of natural numbers. Consider the relation “has its cube as”, from set $N$ to $N$. Thus 1 has its cube as 1; 2 has its cube as 8, 3 as 27, 4 as 64. Writing ‘R’ in place of “has its cube”, we get:

1R1, 2R8, 3R27, 4R64, ...

$\therefore R=\left\{\right(1,1),(2,8),(3,27),(4,64),\dots \}$

i.e., $R=\left\{\right(x,y)\mid x\in N,y\in N,y=x^3\}$

Domain and Range of a Relation

Let A and B be any two non - empty sets and R be a relation from A to B, then the domain of the relation R is the set of all first components of the ordered pairs which belong to R, and the range of the relation R is the set of all second components of the ordered pairs which belong to R.

Thus, Domain of R =  {x : x ∈ A,(x,y) ∈ R for some y ∈ B} and Range of R = {y : y ∈ B, (x,y) ∈ R for some x ∈ A}

The following are the different types of relations that students must learn about:

1. Reflexive Relations Every element is related to itself. For relation R on set A: aRa for all a ∈ A. Example: "equals" (=), "less than or equal to" (≤)
2. Symmetric Relations If a is related to b, then b is related to a. For all a,b ∈ A: if aRb, then bRa. Example: "is married to", "is siblings with". Based on these types of relations in maths, students will get to practice questions in NCERT excercise of class 12 chapter. 
3. Antisymmetric Relations If a is related to b and b is related to a, then a equals b. For all a,b ∈ A: if aRb and bRa, then a = b. Example: "less than or equal to" (≤), "divides" (for positive integers)
4. Transitive Relations If a is related to b and b is related to c, then a is related to c. For all a,b,c ∈ A: if aRb and bRc, then aRc. Example: "less than" (<), "ancestor of"
5. Empty Relation: The empty relation is the relation that contains no ordered pairs. It's denoted as ∅ or { }.
   For any sets A and B, the empty relation R ⊆ A × B where R = ∅, meaning no element of A is related to any element of B.
6. Universal Relation: The universal relation contains all possible ordered pairs between two sets. For sets A and B, the universal relation R = A × B.
7. Identity Relation: The identity relation on a set A consists of all pairs (a,a) where a ∈ A. It's denoted as IA = {(a,a) : a ∈ A}.
8. Inverse Relation: If R is a relation from set A to set B, then the inverse relation R⁻¹ is a relation from B to A defined as: R⁻¹ = {(b,a) : (a,b) ∈ R}
9. Equivalence Relation: An equivalence relation is a relation that is simultaneously reflexive, symmetric, and transitive. It partitions the set into disjoint equivalence classes.Relation R on set A is an equivalence relation if:

• Reflexive: aRa for all a ∈ A
• Symmetric: if aRb, then bRa
• Transitive: if aRb and bRc, then aRc

Do take a look at these as well:

CBSE Class 12 NCERT Notes	NCERT Class 11 Math Notes
NCERT Class 12 Maths Notes for CBSE	Class 12 Physics NCERT Notes

Relations are expressed in the following ways in Maths:

1. Roster form

In this form, a relation is represented by the set of all ordered pairs which belong to the given relation.

e.g., Let A = {1, 2,3,4,5,6} and B = {1, 2,3.....,30} and R be the relation “has its square as” from A to B, then R in the roster form can be written as

R = {(1,1), (2,4), (3,9), (4,16), (5,25)}

2. Set - Builder form

In this form, a relation is represented as {(x,y): x ∈ A, y ∈ B x...y}, where the blank is to be represented by the rule which associates x and y.

e.g., Let A = {1, 2, 3, 4, 5} and B = {3, 5, 6, 7} and R = {(1,3), (3,5), (4,6), (5,7)}, then R in the set - builder form can be written as

R = (x,y): x ∈ A, y ∈ B x is two less than y}

3. Arrow Diagram

In this form, the relation is represented by drawing arrows from the first components to the second components of all ordered pairs which belong to the given relation.

e.g., Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 20} and R be the relation “is square root” from A to B, then R = {(1,1), (2,4), (3,9), (4,16)}.

This relation R from A to B can be represented by the arrow diagram as shown in the adjoining figure.

If two variables x and y are so related that whenever a value is assigned to x, it gives a unique value of y according to some rule, then y is a function of x and is expressed symbolically as

y = f(x), where x is an independent variable and y is the dependent variable.

Let X, Y be two non – empty sets. A subset of X ∈Y is called a function (or mapping) from X to Y if for each x ∈ X, there exists a unique y ∈Y such that (x, y) ∈ f.

It is written as f: X $\to$ Y.

Thus, a subset f of X x Y is called a function from X to Y if:

• For each x ∈ X, there exists y ∈ Y such that (x, y) ∈ f and
• No two different ordered pairs in f have the same first component.

Image and Pre-image

The unique element y of Y is called the image of the element x of X under the function

 f: X $\to$ Y. It is denoted by f(x) i.e., y = f(x). The element x is called a pre-image or an inverse-image of y.

In other words, if f is a function from X to Y and (x, y) ∈ f, then f(x) = y, where y is called the image of x under f and x is called the pre-image of y under f.

The numbers x and y where y is an image of x are also denoted by ordered pair (x, y) or [(x, f(x))] .

Domain and Range

Let f be a function from X to Y. Then the set X is called the domain of the function f and the set Y is called the co - domain of the function f. The set consisting of all the images of the elements of X under the function f is called the range of f. It is denoted by f(X).

Thus, range of f = {f(x) for all x ∈ X}.

A function in math is represented in the following ways:

1. Roster Form

In this form, a function is represented by the set of all ordered pairs which belong to the given function.

e.g., Let X = {0,2,3,4} and Y = {0,3,4,6,7,8,9,11} and f be the function ‘is less than’ from X to Y, then f = {(0,4),(2,6),(3,7),(4,8)}.

2. Builder Form

In this form, the function is represented as {(x,y): x ∈ X, y ∈ Y x...y}, where the blank is to be replaced by the rule which associates x and y.

The function given above can be written as

 f = {(x,y): x ∈ X, y ∈ Y x is 4 less than y}

3. By Formula

In this form, a formula (an algebraic equation) can be used to represent a function

e.g., y = 5x – 1, where x N.

4. Arrow Diagram

In this form, the function is represented by drawing arrows from the first components to the second components of all ordered pairs which belong to the given function.

Those who want to practice Relations and Functions Class 11 chapter must remember the following points. Even enterance examinations like JEE Main and NEET ask questions based on the concept of relations and functions. Direct questions are rare in these exams. While we have already discussed the basics of the chapter, it is important to go through the sub-topics in detail for better performance,

• Let A and B be two non-empty finite sets having m and n elements respectively.
• Number of ordered pairs in A B = mn.
• Number of elements in A B = mn.
• Total number of relations from A to B is 2^mn.
• The numbers x and y where y is an image of x are also denoted by ordered pair (x, y) or .
• Range of f is a subset of Y (co-domain) which may or may not be equal to Y.
• A graph will represent a function if a vertical line meets the graph in at most one point.
• A graph will not represent a function if a vertical line can be drawn to meet the graph in more than one point.