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RELATIONS AND FUNCTIONS.

Top Concepts in Relations

1.Introduction to Relation and no. of relations

●A relation R between two non-empty sets A and B is a subset of their Cartesian product A × B.

●If A = B, then the relation R on A is a subset of A × A.

●The total number of relations from a set consisting of m elements to a set consisting of n elements is 2^mn.

●If (a, b) belongs to R, then a is related to b and is written as ‘a R b’. If (a, b) does not belong to R, then a is not related to b and it is written as

2.Co-domain and Range of a Relation

Let R be a relation from A to B. Then the ‘domain of and the ‘range of Co-domain is either set B or any of its superset or subset containing range of R.

3.Types of Relations

A relation R in a set A is called an empty relation if no element of A is related to any element of A, i.e.,

A relation R in a set A is called a universal relation if each element of A is related to every element of A, i.e., R = A × A.

4.A relation R on a set A is called:

a.Reflexive, if (a, a)

\in

R for every a

\in

b.Symmetric, if (a₁, a₂)

\in

R implies that (a₂, a₁)

\in

R for all a₁, a₂

\in

c.Transitive, if (a₁, a₂)

\in

R and (a₂, a₃)

\in

R implies that (a1, a3)

\in

R for all a₁, a₂, a₃

\in

5.Equivalence Relation

●A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

●An empty relation R on a non-empty set X (i.e., ‘a R b’ is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, but it is not reflexive (except when X is also empty).

6.Given an arbitrary equivalence relation R in a set X, R divides X into mutually disjoint subsets S_i called partitions or subdivisions of X provided:

a.All elements of S, are related to each other for all i.

b.No element of Si is related to any element of St if i

\neq

The subsets St are called equivalence classes.

7.Union, Intersection and Inverse of Equivalence Relations

a.If R and S are two equivalence relations on a set A, R

\cap

S is also an equivalence relation on A.

b.The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

c.The inverse of an equivalence relation is an equivalence relation.

Top Concepts in Functions

1.Introduction to functions

A function from a non-empty set A to another non-empty set B is a correspondence or a rule which associates every element of A to a unique element of B written as f : A → B such that f(x) = y for all x $\in$ A, y $\in$ B.

All functions are relations, but the converse is not true.

2.Domain, Co-domain and Range of a Function

●If f : A → B is a function, then set A is the domain, set B is the co-domain and set {f(x) : x

\in

A) is the range of f.

●The range is a subset of the co-domain.

●A function can also be regarded as a machine which gives a unique output in set B corresponding to each input from set A.

●If A and B are two sets having m and n elements, respectively, then the total number of functions from A to B is n^m.

3.Real Function

●A function f : A → B is called a real-valued function if B is a subset of R.

●If A and B both are subsets of R, then ‘f’ is called a real function.

●While describing real functions using mathematical formula, x (the input) is the independent variable and y (the output) is the dependent variable.

●The graph of a real function ‘f’ consists of points whose co-ordinates (x, y) satisfy y = f(x), for all x

\in

Domain(f).

4.Vertical line test

A curve in a plane represents the graph of a real function if and only if no vertical line intersects it more than once.

5.One–one Function

●A function f : A → B is one-to-one if for all x, y

\in

A, f(x) = f(y) ⇒ x = y or x ≠ y ⇒ f(x) ≠ f(y).

●A one-one function is known as an injection or injective function. Otherwise, f is called many-one.

6.Onto Function

●A function f : A → B is an onto function, if for each b

\in

B, there is at least one a

\in

A such that f(a) = b, i.e., if every element in B is the image of some element in A, then f is an onto or surjective function.

●For an onto function, range = co-domain.

●A function which is both one-one and onto is called a bijective function or a bijection.

●A one-one function defined from a finite set to itself is always onto, but if the set is infinite, then it is not the case.

7.Let A and B be two finite sets and f : A → B be a function.

●If f is an injection, then n (A)

\leq

n (B) .

●If f is a surjection, then n(A)

\geq

n(B).

●If f is a bijection, then n(A) = n(B) .

8.If A and B are two non-empty finite sets containing m and n elements, respectively, then Number of functions from A to B = n^m.

●Number of one-one function from A to B A picture containing text Description automatically generated

●Number of onto functions from A to B A picture containing table Description automatically generated

●Number of one-one and onto functions from A to B

9.If a function f : A → B is not an onto function, then f : A → f(A) is always an onto function.

10.Composition of Functions

Let f : A → B and g : B → C be two functions. The composition of f and g, denoted by g o f, is defined as the function g o f: A → C and is given by g o f(x): A → C defined by g o f(x) = g(f(x)) $\forall x \in$ A.

●Composition of f and g is written as g o f and not f o g.

●g o f is defined if the range of f

domain of g, and f o g is defined if the range of g

domain of f.

●Composition of functions is not commutative in general i.e., f o g(x) ≠ g o f(x).

●Composition is associative i.e., if f : X → Y, g : Y → Z and h : Z → S are functions, then h o (g o f) = (h o g) o f.

●The composition of two bijections is a bijection.

11.Inverse of a Function

●Let f : A → B is a bijection, then g : B → A is inverse of f if f(x) = y

⟺

g(y) = x OR g o f = l_A and f o g = l_B

●If g o f = l_A and f is an injection, then g is a surjection.

●If f o g’ l_B and f is a surjection, then g is an injection.

12. Let f : A → B and g: B → C be two functions. Then

●g o f: A → C is onto ⇒ g: B → C is onto.

●g o f: A → C is one-one ⇒ f:A → B is one-one.

●g o f: A → C is onto and g: B → C is one-one ⇒ f:A → B is onto.

●g o f: A → C is one-one and f:A → B is onto ⇒ g: B → C is one-one.

13. Invertible Function

●A function f : X → Y is defined to be invertible if there exists a function g : Y → X such that gof – l_x and fog = I_y.

●The function g is called the inverse of f and is denoted by f^-1. If f is invertible, then f must be one-one and onto, and conversely, if f is one-one and onto, then f must be invertible.

●If f : A → B and g : B → C are one-one and onto, then g o f : A → C is also one-one and onto. But if g o f is one-one, then only f is one-one and g may or may not be one-one. If g o f is onto, then g is onto and f may or may not be onto.

●Let f : X → Y and g : Y → Z be two invertible functions. Then g o f is also invertible with (g o f)^-1 = f^-1 o g^-1.

●If f: R → R is invertible, f(x) = y, then f^-1 (y) = x and (f^-1)^-1 is the function f itself.

Binary Operations

1.A binary operation * on a set A is a function from A × A to A.

2.If * is a binary operation on a set S, then S is closed with respect to *.

3.Binary operations on R

●Addition, subtraction and multiplication are binary operations on R, which is the set of real numbers.

●Division is not binary on R; however, division is a binary operation on R – {0} which is the set of non-zero real numbers.

4.Laws of Binary Operations

●A binary operation * on the set X is called commutative, if a * b = b * a, for every a, b

\in

●A binary operation * on the set X is called associative, if a (b*c) = (a*b)*c, for every a, b, c

\in

●An element e

\in

A is called an identity of A with respect to * if for each a

\in

A, a * e = a = e * a.

●The identity element of (A, *) if it exists, is unique.

5.Existence of Inverse

Given a binary operation * from A × A → A with the identity element e in A, an element a e A is said to be invertible with respect to the operation *, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a^-1.

6.If the operation table is symmetric about the diagonal line, then the operation is commutative.

Chart, line chart Description automatically generated

The operation * is commutative.

7.Binary Operation on Natural Numbers

Addition ’+’ and multiplication ‘-‘ on N, the set of natural numbers, are binary operations. However, subtraction ‘—’ and division are not, because (4, 5) = 4 – 5 = -1 $\in$ N and 4/5 = .8 $\in$ N.

8.Number of Binary Operations

●Let S be a finite set consisting of n elements. Then S x S has n² elements.

●The total number of functions from a finite set A to a finite set B is [n(B)]ⁿ^(A). Therefore, total number of binary operations on S is

n^{n^{2}}

●The total number of commutative binary operations on a set consisting of n elements is n

\frac{n (n - 1)}{2} .

Important Questions

Multiple Choice questions-

1.Let R be the relation in the set (1, 2, 3, 4}, given by:

R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.

Then:

(a) R is reflexive and symmetric but not transitive

(b) R is reflexive and transitive but not symmetric

(d) R is an equivalence relation.

2. Let R be the relation in the set N given by: R = {(a, b): a = b – 2, b > 6}. Then:

(a) (2, 4) ∈ R

(b) (3, 8) ∈ R

(d) (8, 7) ∈ R.

3. Let A = {1, 2, 3}. Then number of relations containing {1, 2} and {1, 3}, which are reflexive and symmetric but not transitive is:

(a) 1

(b) 2

(d) 4.

4. Let A = (1, 2, 3). Then the number of equivalence relations containing (1, 2) is

(a) 1

(b) 2

(d) 4.

5. Let f: R → R be defined as f(x) = x⁴. Then

(a) f is one-one onto

(b) f is many-one onto

(d) f is neither one-one nor onto.

6. Let f: R → R be defined as f(x) = 3x. Then

(a) f is one-one onto

(b) f is many-one onto

(d) f is neither one-one nor onto.

7. If f: R → R be given by f(x) = (3 – x³)^1/3, then f_of (x) is

(a) x^1/3

(b) x³

(d) 3 – x³.

8. Let f: R – {- $\frac{4}{3}$ } → R be a function defined as: f(x) = $\frac{4 x}{3 x + 4}$ , x ≠ – $\frac{4}{3}$ . The inverse of f is map g: Range f → R -{– $\frac{4}{3}$ } given by

(a) g(y) = $\frac{3 y}{3 - 4 y}$

(b) g(y) = $\frac{4 y}{4 - 3 y}$

(d) g(y) = $\frac{3 y}{4 - 3 y}$

9. Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is

(a) Reflexive and symmetric

(b) Transitive and symmetric

(d) Reflexive, transitive but not symmetric.

10. Set A has 3 elements, and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is:

(a) 144

(b) 12

(d) 64

Very Short Questions:

1.If R = {(x, y) : x + 2y = 8} is a relation in N, write the range of R.

2.Show that a one-one function:

f {1, 2, 3} → {1, 2, 3} must be onto. (N.C.E.R.T.)

3.What is the range of the function f(x) =

\frac{| x - 1 |}{x - 1}

? (C.B.S.E. 2010)

4.Show that the function f : N → N given by f(x) = 2x is one-one but not onto. (N.C.E.R.T.)

5.If f : R → R is defined by f(x) = 3x + 2 find f(f(x)). C.B.S.E. 2011 (F))

6.If f(x) =

\frac{x}{x - 1}

, x ≠1 then find fof. (N.C.E.R.T)

7.If f: R → R is defined by f(x) = (3 – x³)^1/3, find fof (x)

8.Are f and q both necessarily onto, if gof is onto? (N.C.E.R.T.)

Short Questions:

1.Let A be the set of all students of a Boys’ school. Show that the relation R in A given by:

R = {(a, b): a is sister of b} is an empty relation and the relation R’ given by :

R’ = {(a, b) : the difference between heights of a and b is less than 3 metres} is an universal relation. (N.C.E.R.T.)

2.Let f : X → Y be a function. Define a relation R in X given by :

R = {(a,b):f(a) = f(b)}.

Examine, if R is an equivalence relation. (N.C.E.R.T.)

3.Let R be the relation in the set Z of integers given by:

R = {(a, b): 2 divides a – b}.

Show that the relation R is transitive. Write the equivalence class [0]. (C.B.S.E. Sample Paper 2019-20)

4.Show that the function:

f : N → N

given by f(1) = f(2) = 1 and f(x) = x -1, for every x > 2 is onto but not one-one. (N.C.E.R.T.)

5.Find gof and fog, if:

f : R → R and g : R → R are given by f (x) = cos x and g (x) = 3x². Show that gof ≠ fog. (N. C.E.R. T.)

6.If f(x) =

\frac{4 x + 3}{6 x - 4}

, x ≠

\frac{2}{3}

find fof(x)

7.Let A = N x N be the set of ail ordered pairs of natural numbers and R be the relation on the set A defined by (a, b) R (c, d) iff ad = bc. Show that R is an equivalence relation.

8.Let f: R → R be the Signum function defined as:

Class 12 Maths Important Questions Chapter 1 Relations and Functions 3

and g : R → R be the Greatest Integer Function given by g (x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1]?

Long Questions:

1.Show that the relation R on R defined as R = {(a, b):a ≤ b}, is reflexive and transitive but not symmetric.

2.Prove that function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find inverse of f : N → S, where S is range of f.

3.Let A = (x ∈Z : 0 ≤ x ≤ 12}.

Show that R = {(a, b) : a, b ∈ A; |a – b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]. (C.B.S.E 2018)

4.Prove that the function f: [0, ∞) → R given by f(x) = 9x² + 6x – 5 is not invertible. Modify the co-domain of the function f to make it invertible, and hence find f-1. (C.B.S.E. Sample Paper 2018-19

Assertion and Reason Questions-

1. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes(a), (b), (c) and (d) as given below.

a)Both A and R are true and R is the correct explanation of A.

b)Both A and R are true but R is not the correct explanation of A.

c)A is true but R is false.

d)A is false and R is also false.

Assertion(A): Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2): L1 is perpendicular to L2}.R is not equivalence realtion.

Reason (R): R is symmetric but neither reflexive nor transitive

2. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes(a), (b), (c) and (d) as given below.

a)Both A and R are true and R is the correct explanation of A.

b)Both A and R are true but R is not the correct explanation of A.

c)A is true but R is false.

d)A is false and R is also false.

Assertion (A): = {(T1, T2): T1 is congruent to T2}. Then R is an equivalence relation.

Reason(R): Any relation R is an equivalence relation, if it is reflexive, symmetric and transitive.

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