Relations and Functions Mathematics Class 12

EXERCISE 1.1 

 1. Determine whether each of the following relations are reflexive, symmetric andtransitive:(i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined asR = {(x, y) : 3x – y = 0}(ii) Relation R in the set N of natural numbers defined asR = {(x, y) : y = x + 5 and x < 4}(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} asR = {(x, y) : y is divisible by x}(iv) Relation R in the set Z of all integers defined asR = {(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}

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

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

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

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

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

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 equivalencerelation.

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

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.

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.

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

12. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1is similar to T2}, is equivalence relation. Consider three right angle triangles T1with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Whichtriangles among T1, T2 and T3 are related?

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 theset of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

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

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.

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

EXERCISE 1.2 

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

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

3. Prove that the Greatest Integer Function f : R→R, given by f (x) = [x], is neitherone-one nor onto, where [x] denotes the greatest integer less than or equal to x.

4. Show that the Modulus Function f : R→R, given by f (x) = | x |, is neither oneonenor onto, where | x | is x, if x is positive or 0 and | x | is – x, if x is negative.

5. Show that the Signum Function f : R→R, given by1, if 0( ) 0,if 0–1, if 0xf x xx> ⎧⎪= = ⎨⎪⎩ <is neither one-one nor onto.

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

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

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

9. Let f : N → N be defined by f (n) =1, if is odd2, if is even2n nn n+ ⎧⎪⎪⎨⎪⎪⎩for all n ∈ N.State whether the function f is bijective. Justify your answer.

10. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined byf (x) =23xx⎛ − ⎞⎜ ⎟⎝ − ⎠ . Is f one-one and onto? Justify your answer.

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

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 

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

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

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

4. If f (x) =(4 3)(6 4)xx+− ,23x ≠ , show that fof (x) = x, for all 23x ≠ . What is theinverse of f ?

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

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

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

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

9. Consider f : R+→ [– 5, ∞) given by f (x) = 9x2 + 6x – 5. Show that f is invertiblewith f –1(y)=( 6 ) 13⎛ y + − ⎞⎜ ⎟⎝ ⎠.

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).

11. Consider f : {1, 2, 3} → {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Findf –1 and show that (f –1)–1 = f.12. Let f : X → Y be an invertible function. Show that the inverse of f –1 is f, i.e.,(f –1)–1 = f.

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

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

14. Let f : R – 43⎧− ⎫ ⎨ ⎬⎩ ⎭→ R be a function defined as f (x) =43 4xx +. The inverse off is the map g : Range f → R – 43⎧− ⎫ ⎨ ⎬⎩ ⎭given by(A)( ) 33 4g y yy=− (B)( ) 44 3g y yy=−(C)( ) 43 4g y yy=− (D)( ) 34 3g y yy=−

EXERCISE 1.4 

1. Determine whether or not each of the definition of ∗ given below gives a binaryoperation. 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

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

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

4. Consider a binary operation ∗ on the set {1, 2, 3, 4, 5} given by the followingmultiplication 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)Table 1.2

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

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 ∗?

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

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 binaryoperation on N?

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 =4ab(vi) a ∗ b = ab2Find which of the binary operations are commutative and which are associative.

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

11. 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 ∗ onA, if any.

12. 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

13. Consider a binary operation ∗ on N defined as a ∗ b = a3 + b3. Choose thecorrect 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?

Miscellaneous Exercise on Chapter 1 

1. Let f : R → R be defined as f (x) = 10x + 7. Find the function g : R → R suchthat g o f = f o g = 1R.

2. Let f : W → W be defined as f (n) = n – 1, if n is odd and f (n) = n + 1, if n iseven. Show that f is invertible. Find the inverse of f. Here, W is the set of allwhole numbers.

3. If f : R → R is defined by f(x) = x2 – 3x + 2, find f (f (x)).

4. Show that the function f : R→ {x ∈ R : – 1 < x < 1} defined by ( )1 | |f x xx=+ ,x ∈ R is one one and onto function.

5. Show that the function f : R →R given by f (x) = x3 is injective.

6. Give examples of two functions f : N → Z and g : Z → Z such that g o f isinjective but g is not injective.(Hint : Consider f (x) = x and g (x) = | x |).

7. Give examples of two functions f : N→N and g : N→N such that g o f is ontobut f is not onto.(Hint : Consider f (x) = x + 1 and1if 1( )1 if 1x xg xx⎧ − >= ⎨ = ⎩

8. Given a non empty set X, consider P(X) which is the set of all subsets of X.Define the relation R in P(X) as follows:For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relationon P(X)? Justify your answer.

9. Given a non-empty set X, consider the binary operation ∗ : P(X) × P(X) → P(X)given by A ∗ B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X.Show that X is the identity element for this operation and X is the only invertibleelement in P(X) with respect to the operation ∗.

10. Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.

11. Let S = {a, b, c} and T = {1, 2, 3}. Find F–1 of the following functions F from Sto T, if it exists.(i) F = {(a, 3), (b, 2), (c, 1)} (ii) F = {(a, 2), (b, 1), (c, 1)}

12. Consider the binary operations ∗ : R × R → R and o : R × R → R defined asa ∗b = |a – b| and a o b = a, ∀ a, b ∈ R. Show that ∗ is commutative but notassociative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R,a ∗ (b o c) = (a ∗ b) o (a ∗ b). [If it is so, we say that the operation ∗ distributesover the operation o]. Does o distribute over ∗? Justify your answer.

13. Given a non-empty set X, let ∗ : P(X) × P(X) → P(X) be defined asA * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is theidentity for the operation ∗ and all the elements A of P(X) are invertible withA–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).

14. Define a binary operation ∗ on the set {0, 1, 2, 3, 4, 5} as, if 66 if 6a b a ba ba b a b⎧ + + <∗ = ⎨ + − + ≥ ⎩Show that zero is the identity for this operation and each element a of the set isinvertible with 6 – a being the inverse of a.

15. Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions definedby f (x) = x2 – x, x ∈ A and( ) 2 1 1,2g x = x − − x ∈ A. Are f and g equal?Justify your answer. (Hint: One may note that two functions f : A → B andg : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions).

16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which arereflexive and symmetric but not transitive is(A) 1 (B) 2 (C) 3 (D) 4

17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is(A) 1 (B) 2 (C) 3 (D) 4

18. Let f : R→ R be the Signum Function defined as1, 0( ) 0, 01, 0xf x xx> ⎧⎪= = ⎨⎪⎩− <and g : R → R be the Greatest Integer Function given by g (x) = [x], where [x] isgreatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1]?

19. Number of binary operations on the set {a, b} are(A) 10 (B) 16 (C) 20 (D ) 8