Class 12 Maths NCERT Solutions For Chapter 1 Relations and Functions

Chapter 1 Relations and Functions

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In this pdf file you can see answers of following Questions

EXERCISE 1.1

Question 1. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}
(iv) Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}
(v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {(x, y) : x and y work at the same place} (b) R = {(x, y) : x and y live in the same locality} (c) R = {(x, y) : x is exactly 7 cm taller than y} (d) R = {(x, y) : x is wife of y} (e) R = {(x, y) : x is father of y}

Question 2. Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.

Question 3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Question 4. Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Question 5. Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.

Question 6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Question 7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

Question 8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Question 9. Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by
(i) R = {(a, b) : |a – b| is a multiple of 4}
(ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Question 10. Give an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Question 11. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Question 12. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8,Which triangles among T1, T2 and T3 are related?

Question 13.Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 14.Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x +4

Question 15.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)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Question 16. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer.
(A) (2, 4) ∈ R
(B) (3, 8) ∈ R
(C) (6, 8) ∈ R
(D) (8, 7) ∈ R

EXERCISE 1.2

Question 1.Show that the function f : R → R defined by f (x) = 1 x is one-one and onto, where R is the set of all non-zero real numbers. Is the result true, if the domain R is replaced by N with co-domain being same as R?

Question 2.Check the injectivity and surjectivity of the following functions:
(i) f : N → N given by f (x) = x2
(ii) f : Z → Z given by f (x) = x2
(iii) f : R → R given by f (x) = x2
(iv) f : N → N given by f (x) = x3 (v) f : Z → Z given by f (x) = x3

Question 3.Prove that the Greatest Integer Function f : R→R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Question 4.Show that the Modulus Function f : R→R, given by f (x) = | x |, is neither oneone nor onto, where | x | is x, if x is positive or 0 and | x | is – x, if x is negative.

Question 5.Show that the Signum Function f : R→R, given by 1, if 0 ( ) 0,if 0 –1, if 0 x f x x x > = = is neither one-one nor onto.

Question 6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

Question 7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f (x) = 3 – 4x
(ii) f : R → R defined by f (x) = 1 + x2

Question 8. Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.

Question 9. Let f : N → N be defined by f (n) = 1, if is odd 2 , if is even 2 n n n n + for all n ∈ N. State whether the function f is bijective. Justify your answer.

Question 10. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f (x) = 2 3 x x − − . Is f one-one and onto? Justify your answer.

Question 11.Let f : R → R be defined as f(x) = x 4.3Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.

Question 12.Let f : R → R be defined as f (x) = 3x. Choose the correct answer. (A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.

EXERCISE 1.3

Question 1.Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Question 2.Let f, g and h be functions from R to R. Show that (f + g)oh = foh + goh (f . g)oh = (foh) . (goh)

Question 3.Find gof and fog, if (i) f (x) = | x | and g(x) = | 5x – 2 | (ii) f (x) = 8x3 and g(x) = 1 x 3.

Question 4.If f (x) = (4 3) (6 4) x x + − , 2 3 x ≠ , show that fof (x) = x, for all 2 3 x ≠ . What is the inverse of f ?

Question 5.State with reason whether following functions have inverse
(i) f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Question 6. Show that f : [–1, 1] →R, given by f (x) = ( 2) x x + is one-one. Find the inverse of the function f : [–1, 1] → Range f. (Hint: For y ∈ Range f, y = f (x) = 2 x x + , for some x in [–1, 1], i.e., x = 2 (1 ) y − y )

Question 7. Consider f : R → R given by f (x) = 4x + 3.Show that f is invertible. Find the inverse of f.

Question 8. Consider f : R+→ [4, ∞) given by f (x) = x2 +4..Show that f is invertible with the inverse f –1 of f given by f –1(y) = y − 4 , where R+ is the set of all non-negative real numbers.

Question 9. Consider f : R+→ [– 5, ∞) given by f (x) = 9x2 + 6x – 5. Show that f is invertible with f –1(y)= ( 6 ) 1 3 y + − .

Question 10. Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).

Question 11.Consider f : {1, 2, 3} → {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Find f –1 and show that (f –1)–1 = f.

Question 12.Let f : X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f –1)–1 = f.

Question 13.If f : R → R be given by f (x) = 1 (3 − x3 )3 , then fof (x) is (A) 1 x3 (B) x3 (C) x (D) (3 – x3)

Question 14.Let f : R – 4 3 ⎧− → R be a function defined as f (x) = 4 3 4 x x + . The inverse of f is the map g : Range f → R – 4 3 − given by
(A) ( ) 3 3 4 g y y y = −
(B) ( ) 4 4 3 g y y y = −
(C) ( ) 4 3 4 g y y y = −
(D) ( ) 3 4 3 g y y y

EXERCISE 1.4

Question 1.Determine whether or not each of the definition of given below gives a binary operation. In the event that is not a binary operation, give justification for this.
(i) On Z+, define by a b = a – b
(ii) On Z+, define by a b = ab
(iii) On R, define by a b = ab2 (iv) On Z+, define by a b = | a – b | (v) On Z+, define by a b = a

Question 2.For each binary operation defined below, determine whether is commutative or associative.
(i) On Z, define a b = a – b
(ii) On Q, define a b = ab + 1
(iii) On Q, define a b = 2 ab
(iv) On Z+, define a b = 2ab
(v) On Z+, define a b = ab
(vi) On R – {– 1}, define a b = 1 a b +

Question 3.Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧ .

Question 4.Consider a binary operation on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).
(i) Compute (2 3) 4 and 2 (3 4)
(ii) Is commutative?
(iii) Compute (2 3) (4 5). (Hint: use the following table)

Question 5.Let ′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a ′ b = H.C.F. of a and b. Is the operation ′ same as the operation defined in Exercise 4 above? Justify your answer.

Question 6. Let be the binary operation on N given by a b = L.C.M. of a and b. Find
(i) 5 7, 20 16
(ii) Is commutative?
(iii) Is associative?
(iv) Find the identity of in N
(v) Which elements of N are invertible for the operation ?

Question 7. Is defined on the set {1, 2, 3, 4, 5} by a b = L.C.M. of a and b a binary operation? Justify your answer.

Question 8. Let be the binary operation on N defined by a b = H.C.F. of a and b. Is commutative? Is associative? Does there exist identity for this binary operation on N?

Question 9. Let be a binary operation on the set Q of rational numbers as follows:
(i) a b = a – b
(ii) a b = a2 + b2
(iii) a b = a + ab
(iv) a b = (a – b)2 (v) a b = 4 ab
(vi) a b = ab2 Find which of the binary operations are commutative and which are associative.

Question 10. Show that none of the operations given above has identity.

Question 1.Let A = N × N and be the binary operation on A defined by (a, b) (c, d) = (a + c, b + d)
Show that is commutative and associative. Find the identity element for on A, if any.

Question 2.State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation on a set N, a a = a ∀ a ∈ N.
(ii) If is a commutative binary operation on N, then a (b c) = (c b) a

Question 3.Consider a binary operation on N defined as a b = a3 + b3.Choose the correct answer.
(A) Is both associative and commutative?
(B) Is commutative but not associative?
(C) Is associative but not commutative?
(D) Is neither commutative nor associative?

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