RELATIONS, FUNCTIONS AND OPERATIONS.pdf

RELATIONS, FUNCTIONS, OPERATIONS 1 BY SUDHIR SAXENA

GYAN BHARATI SCHOOL, SAKET, N.D. MATHEMATICS ASSIGNMENT : CLASS SS2

RELATIONS, FUNCTIONS, OPERATIONS Date of Uploading : 27/4/2015 ; Date of Submission : 18/5/2015

PREVIOUS YEAR'S QUESTIONS :

Q1(i ) Is the binary operat ion def ined on set N, giv en by a b = for al l a, b N, commutativ e?

( i i ) Is the abov e binary operat ion associat iv e? (2008, 2011)

Q2 If f (x ) = x + 7 and g(x ) = x 7, x R, f ind (fog)(7). (2008)

Q3 Is f (x ) an inv ert ible funct ion? Also f ind the inverse of f (x ) , i f possible. f (x ) = . (AI2008)

Q4 Let T be the set of al l t riangles in a plane wi th R as relat ion in T given by

R = {(T 1 , T2) : T 1 T2} . Show that R is equiv alence relat ion. (AI2008) Q5 Show that the relat ion R in the set A = {1, 2, 3, 4, 5} given by

R = {(a, b) : |a b| is ev en}, is an equiv alence relat ion. Show that al l the elements of {1, 3, 5} are related to each other and al l 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}. (2009)

Q6 If the binary operat ion on the set of integers Z, is def ined by a b = a + 3b2, then f ind the

value of 2 4. (2009)

Q7 Let be a binary operat ion on N given by a b = HCF of a and b where

a, b N. W ri te the value of 22 4. (AI2009) Q8 Let f : N N be def ined by

i f n is odd

f (n) =

i f n is ev en

for al l n N. Find if the funct ion is bi ject iv e or not ? (AI2009)

Q9 If the binary operat ion def ined on Q, is def ined as a b = 2a + b ab for al l a, b Q, f ind the value of 3 4. (F2009)

Q10 Show that the relat ion R on the set of the real numbers, def ined as

R = {(a, b) : a b2} is nei ther ref lex ive, nor symmetric, nor t ransi t iv e. (F2009)

Q11 Let Z be the set of al l integers and R be relat ion on Z def ined as

R = {(a, b) : a, b Z and a b is div isible by 5}. Prov e that R is an equiv alence relat ion. (2010)

Q12 What is the range of the funct ion f (x ) = . (2010)

Q13 Show that the relat ion R on the set A = {x Z : 0 x 12}, giv en by R = {a, b) : |a b| is a mul tiple of 4} is an equiv alence relat ion. (AI2010)

Q14 If f : R R giv en by f (x ) = (3 x

3)

1 / 3, determine f (f (x )) . (AI2010)

GYAN BHARATI SCHOOL, SAKET, NEW DELHI


Q15 Is f (x ) an inv ert ible funct ion? f (x ) = . Also wr i te f- 1

(x ), i f ex ist ing. (F2010)

Q16 Let be def ined on N x N by (a, b) (c, d) = (a + c, b + d). Show that i t is commutat ive as wel l

as associat iv e. Also f ind ident i ty element for on A, i f any. (F2010) Q17 Consider the Binary operat ion on the set {1, 2, 3, 4, 5} def ined by a b = min. {a, b}. Wri te

i ts operat ion table. (2011) Q18 State the reason why R = {(1, 2), (2, 1)}, def ined on the set {1, 2, 3} , is not t ransi t iv e. (2011)

Q19 A binary operat ion on A = {0, 1, 2, 3, 4, 5} is def ined as a b = , show

that 0 is the ident i ty element of this operat ion and each element a of the set is inv ert ible wi th 6 a being the inv erse of a. (AI2011)

Q20 Let f : R R be def ined as f (x ) = 10x + 7. Find the function g : R R such that gof = fog = I R . (AI2011) Q21 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a funct ion f rom A to B.

State whether f is one-one or not . Giv e reason. (AI2011) Q22 If f : R R is def ined by f (x ) = 3x + 2, def ine f ( f (x )) . (F2011) Q23 Consider f : R+ [4, ) giv en by f (x ) = x

2 + 4. Show that f is inv ert ible wi th the inv erse giv en

by f- 1

= , where R+ is the set of al l non negat iv e real numbers. (F2011, 2013) Q24 Wri te fog, i f f : R R and g : R R are giv en by f (x ) = 8x

3 and g(x ) = x

1 / 3. (F2011)

Q25 Let A = R {3} and B = R {1}. Consider the funct ion f : A B def ined by f (x ) = . Show

that f is one-one and onto and hence f ind f- 1

. (2012)

Q26 Let be a binary operat ion on N giv en by a b = LCM(a, b) for al l a, b N. Find 5 7. (2012) Q27 Consider the Binary operat ion : R x R R and o : R x R R def ined as a b = |a b| and

a o b = a for al l a, b R. Show that is commutativ e but not associat iv e, o is associat ive but not commutat ive. (AI2012)

Q28 The Binary operat ion : R x R R is def ined as a b = 2a + b. Find (2 3) 4. (AI2012) Q29 If the binary operat ion on the set of integers Z is def ined by a b = a + b 5, then wri te the

ident i ty element for the operat ion. (F2012)

Q30 Let f : N N be def ined by , show that f is one-one and onto. (AI2012)

Q31 If f (x ) = , x , show that fof (x ) = x for al l x . What is the inv erse of f ? (F2012, 2013)

Q32 Prov e that the relat ion R in the set A = {5, 6 , 7, 8, 9} giv en by R = {(a, b) : |a b| , is div isible

by 2}, is an equiv alence relat ion. Find All elements related to the element 6. (F2013)

Q33 Show t that the function f in A = R def ined as f (x ) = is one-onne and onto. Hence f ind

f1. (2013)



Q34 Let be a binary operat ion, on the set of al l non -zero real numbers, giv en by a b = for al l

a, b R {0}. F ind the v alue of x , giv en that 2 (x 5) = 10. (2014) Q35 Let A = {1, 2, 3, ---- ,9} and R be a relat ion in A x A def ined by (a, b) R (c, d) i f a + d = b + c for

al l (a, b), (c, d) in A x A. Prov e that R is an equiv alence relation. Also obtain the equivalence class [ (2, 5)] . (2014)

Note : I f R is an equiv alence relat ion on A, then equiv alence class of a A is the set of al l elements of

A, which are related to a. Equiv alence class of a is represennted by [a] . Q36 Let N denote the set of al l natural numbers and R be the relat ion on NxN def ined by (a,b)R(c,d)

i f ad(b + c) = bc(a + d). Show that R is an equiv alence relat ion. (2015)

EXTRA PRACTICE QUESTIONS :

1. Def ine relation, funct ion and binary operat ion. 2. How many relat ions are possible f rom A B if n(A) = p and n(B) = q ? How many of them are ( i )

inject ions, ( i i ) sur ject ions, and , ( i i i ) bi jections. 3. Give necessary condi t ions for a relat ion to be a funct ion. In Q2 how many relat ions ar e

funct ions? Also f ind the number of binary operat ions def ined on A and B in Q2. 4. Prov e that a relat ion R def ined on R (the set of real numbers) as fol lows: ( i ) R = {(a, b) : a b}, is not symmetric. ( i i ) R = {(a, b) : a b

2} , is nei ther ref lexiv e, nor symmetric, nor t ransi t iv e.

( i i i ) R = {(a, b) : a b

3} , is nei ther ref lexiv e, nor symmetric, nor t ransi t iv e.

5. Show that the relat ion R on W (set of whole numbers) def ined as R = {(a, b) : |a b| is a mul tiple of 5}, is an equiv alence relat ion. 6. Show that the fol lowing relat ions def ined on N x N are equiv alence relat ions:

( i ) (a, b) R (c, d) ad(b + c) = bc(a + d)

( i i ) (a, b) R (c, d) a + d = b + c

( i i i ) (a, b) R (c, d) ad = bc for al l (a, b), (c, d) N x N. 7. Find the relat ion R def ined on A = {1, 2, 3, 4, 5} by R = {(a, b) : |a

2 b

2| < 16}. Also f ind

domain, range and co domain of R. 8. Def ine and draw graphs of ( i ) greatest integer function, ( i i ) signum funct ion.

9. If f : R R be giv en by f (x ) = sin2x + sin

2(x + /3) + cosx cos(x + /3) for al l

x R and g : R R be such that g(5/4) = 1, prov e that gof i s a constant funct ion. 10. Check fol lowing funct ions for injection, sur ject ion and bi ject ion: ( i ) f : N N ; f (x ) = x

2 + 2x + 3.



( i i ) f : Q {3} Q ; f (x ) = .

( i i i ) f : R R ; f (x ) = 3x + 5.

( iv ) f : [0, ) [7, ) ; f (x ) = 25x2 + 10x 7.

(v ) f : R {x R : 1 < x < 1} ; f (x ) = .

(v i ) f : N x N ; f (n) = n (1)

n.

(v i i ) f : N {0} N {0} ; n + 1 i f n is ev en

f (n) = n 1 i f n is odd (v i i i ) f : R R ; f (x ) = x [x ] where [ . ] represents greatest integer funct ion. ( ix ) f : R R ; f (x ) = 2x

3 + 3.

(x ) f : R R ; f (x ) = x

3 x .

(xi ) f : R R ; f (x ) = x

2 + 2.

11. Which of the fol lowing graphs represent funct ions? Just i f y your answer. ( i ) ( i i ) ( iv ) ( i i i ) 12. Which of the fol lowing funct ions are one one and which are manyone? Also decide which are onto and which are into funct ionns? (Each funct ion is f rom R to R) ( i ) ( i i )



( i i i ) ( iv )

13. Let f : Z Z be def ined by f (n) = 3n n Z and g : Z Z be def ined by: n/3 i f n is a mul tiple of 3

g(n) = n Z 0 i f n is not a mul tiple of 3 Show that gof = I z and fog I z . 14. Let f : R R be signum funct ion and g : R R be greatest integer funct ion, then prov e that fog

and gof coincide in [1, 0). 15. If f (x ) = e

x and g(x ) = logx (x > 0), f ind fog and gof . Are they equal?

16. Let f (x ) = [x ] and g(x ) = |x | . F ind ( i ) (gof )( 5/3) ( fog)(5/3), and,

( i i ) (gof )(5/3) (fog)(5/3).

17. Let f (x ) = . Show that ( fofof )(x ) = .

18. Prov e that f : R ( -1, 1) def ined by f (x ) = , is inv ertible funct ion . Also f ind f- 1

(x ).

19. Prov e that f : R (0, 2) def ined by f (x ) = + 1, is inv ert ible function. Also f ind f- 1

(x ).

20. Explain wi th reason whether fol lowing funct ions hav e inv erse or not : ( i ) f : {1, 2, 3, 4} {10} ; f = {(1, 10), (2, 10), (3, 10), (4, 10)} ( i i ) g : {5, 6, 7, 8} {1, 2, 3, 4} ; g = {(5, 4), (6, 3), (7, 4), (8, 2)} ( i i i ) f : {2, 3, 4, 5} {7, 9, 11, 13} ; f = {(2, 7), (3, 9), (4, 11), (5, 13)}

21. Find the v alues of for which the function f (x ) = 1 + x , 0 is the inv erse of i tself . 22. Let be the binary operat ion on R {1} def ined as a b = a + b + ab. Find the ident i ty element

and the inv erse of al l inv ertible elements w.r. t . , . 23. Let S = N x N and be an operat ion def ined on S by (a, b) (c, d) = (ad + bc, bd) for al l (a, b),

(c, d) S. Show that ( i ) is commutat ive and associat iv e on S, and, ( i i ) ident i ty element w.r. t . , does not ex ist .



24. Determine which of the fol lowing operat ions are binary: ( i ) on R giv en by a b = a

b.

( i i ) on P = { : a, b R {0}} giv en by A B = AB.

( i i i ) on Z giv en by a b = a/b. ( iv ) on A = {1, 2, 3, 4, 5} giv en by a b = lcm of a and b. 25. Find the number of binary operat ions def ined on A i f ( i ) A = {1, 2, 3},

( i i ) A = {a, b}, ( i i i ) A = { , , , } . 26. Check fol lowing operat ions for commutativ e and associat iv e laws: ( i ) on A = N x N def ined by (a, b) (c, d) = (ac, bd),

( i i ) on R {1} def ined by a b = ,

( i i i ) on Q def ined by a b = (a b)

3,

( iv ) on Z def ined by a b = a + 3b 4. 27. Find the number of commutative binary operat ions def ined on a set of 2 elements. 28. If f (x ) and g(x ) be real v alued funct ions such that ( fog)(x ) = cosx

3 and

(gof )(x ) = cos3x . Find the funct ions f (x ) and g(x ).

29. Let A = {1, 2, 3, 4, 5, 6}. W hich of the fol lowing represents part i t ions giv ing r ise to an

equiv alence relat ion? Explain. ( i ) A1 = {1, 3, 5}, A 2 = {2}, A3 = {4} ( i i ) B1 = {1, 2, 5}, B 2 = {3}, B3 = {4, 6} 30. If f unct ions f and g be def ined by f = {(1, 2), (3, 5), (4, 1), (2, 6)},

g = {(2, 6), (5, 4), (1, 3), (6, 1)}, f ind fog and gof . Also wri te the condi t ion(s) under which fog and gof ex ist .

31. If is def ined on R by a b = , f ind ident i ty element in R, i f any. Find al l inv ertible elements along wi th thei r inv erses, i f any.

32. On the set M = A(x) = { : x R}, of 2 x 2 matrices, f ind the ident i ty element for the

mult ipl ication of matr ices as a binary operat ion. Also f ind al l inv ert ibl e elements along wi th thei r inv erses.

33. Let X be a non empty set and let be a binary operat ion on P(X) def ined by A B = A B A, B P(X). Prov e that is commutativ e as wel l as associat iv e on P(X). Find ident i ty element w.r. t . on P(X). Also show that P(X) is the only inv ertible element of P(X).

34. Let X be a non empty set and be a binary operat ion on P(X) def ined by A B = A B. Find ident i ty element and al l invert ible elements along wi th thei r inv erses.

35. Find the number of equiv alence relat ions on the set {1, 2, 3} containing (1, 2) and (2, 1).



36. Find the number of relat ions on A = {1, 2, 3} containing (1, 2) and (1, 3), which are ref lex ive and symmetr ic but not t ransi t ive.

37. Let A = {1, 2, 3}. Then f ind the number of equivalence relat ions containing

(1, 2). 38. Find the number of binary operat ions on {1, 2} hav ing 1 as ident i ty element and 2 as inverse of

2.

ANSWERS

PREVIOUS YEAR'S QUESTIONS : A1 Commutat ive but not associat iv e A2 7 A3 (5x + 2)/7 A6 50 A7 2 A8 Not bi ject ive

A9 2 A12 { 1, 1} A14 x A15 (5x + 4)/3 A16 No ident i ty element A17

1 2 3 4 5

1 1 1 1 1 1

2 1 2 2 2 2

3 1 2 3 3 3

4 1 2 3 4 4

5 1 2 3 4 5

A18 Since (1, 1) R A20

A21 Yes because each element of A has unique image in B. A22 9x + 8 A24 8x A26 35

A28 18 A29 5 A31

A34 Inv erse = A35 25

A36 [ (2, 5)] = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}

EXTRA PRACTICE QUESTIONS :

2 Relat ions = 2p q

, Inject ions = P(q, p) i f q p and 0 i f q < p,

Surject ions = q r

qC r r

p i f p q and 0 i f p < q, Bi jections = p i f p = q and 0 otherwise.

3. Funct ions = qp, operat ions on A = p

p 2

, operat ions on B = qq 2

7. R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (4, 5), (2, 1),

(3, 1), (4, 1), (3, 2), (4, 2), (4, 3), (5, 4)} 10. ( i ) Injection ; (i i ) Injection ; (i i i ) Bi jection ; (iv ) Bi jection ; (v ) Bijection ;



(v i ) Bi ject ion ; (v i i ) Bi ject ion ; (v i i i ) None ; ( ix ) None ; (x ) Surjection ; (xi ) None 11. 2, 4 12. only 4

t h is oneone and rest are

many-one. ( i i ) and ( iv ) are onto and other two are into funct ions.

15. fog = x = gof 16.( i ) 1 ; ( i i ) 0 18. log1 0

19. loge

20.( i ) No , Not Inject ion ; (i i ) No , Not Injection

( i i i ) Yes 21. = -1 22. Ident i ty Element = 0,

Inv erse of a = 24. ( i ) No ; ( i i ) Yes ; ( i i i ) No ; ( iv ) No

25. 3

9, 16, 4

1 6 26.( i ) Both Commutat ive and associat iv e

( i i ) , ( i i i ), ( iv ) Nei ther commutativ e nor associat iv e 27. 2 28. cosx , x

3 29. 2

30. fog is not def ined, gof = {(1, 6), (3, 4), (4, 3), (2, 1)} . fog wi l l exist i f range of g(x ) is a subset of

domain of f (x ). gof wi l l ex ist i f range of f (x ) is a subset of domain of g(x ). 31. Ident i ty element is 0 ; 0 is

the only inv ertible element and i ts inv erse is also 0.

32. Ident i ty element is A( ) ; A( ) is the inv erse of A(x), x 0

33. Ident i ty element = 34. Ident i ty element = X ; X is the only inv ertible element and i ts inv erse is also X. 35. 2 36. 1 37. 2 38. 1

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RELATIONS, FUNCTIONS AND OPERATIONS.pdf