### Exercise 1.1

**Question 1. Determine whether each of the following relations are reflexive, symmetric and transitive:**

**(v)** R is a set of human beings in a town at a particular time.

**(a)** R = {(x, y) : x and y work at the same place}.

Reflexive : It is reflexive as x works at the same place.

Symmetric : It is symmetric since x and y or y and x work at same

place.

Transitive : If x, y work at the same place and y, z work at the same

place, then x and z also work at the same place. Þ it is transitive**(b)** R : {(x, y) : x and y live in the same locality}

With similar reasoning as in part (a), R is reflexive, symmetric and

transitive.**(c)** R : {(x, y) : x is exactly 7 cm taller than y}

Reflexive : it is not reflexive since x cannot be 7 cm taller than x. (x & x

are of the same height).

Symmetric : It is not symmetric because if x is exactly 7 cm taller than

y then y cannot be exactly 7 cm taller than x as y will be exactly 7 cm

shorter than x.

Transitive : It is not transitive because if x is exactly 7 cm taller than

y and if y is exactly 7 cm taller than z, then x is exactly 14 cm taller than

z not 7 cm.

**(d)** R = {(x, y) : x is wife of y}

Reflexive : R is not reflexive : x cannot be wife of x.

Symmetric : R is not symmetric : x is wife of y but y is not wife of x.

Transitive : R is not transitive : if x is a wife of y then y cannot be the

wife of anybody else.**(e)** R = {(x, y) : x is a father of y}

Reflexive : It is not reflexive : x cannot be father of himself.

Symmetric : It is not symmetric : x is a father of y but y cannot be the

father of x.

Transitive : It is not transitive : x is a father of y and y is a father of z

then x cannot be the father of z.

**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} asR = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.**

**Sol.** R = {(a, b) : b = a + 1} in the set {1, 2, 3, 4, 5, 6}.

R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}.**(i)** Q (1, 1) Ï R Þ R is not reflexive.**(ii)** Q (1, 2) Î R but (2, 1) Ï R. Þ R is not symmetric.**(iii)** Q (1, 2), (2, 3) Î R, but (1, 3) Ï R. Þ R is not transitive.

**Question 3. Show that the relation R in R defined as R = {(a, b) : a £ b}, is reflexive and transitive but not symmetric.**

**Sol.** R = {(a, b) : a £ b}**(i)** R is reflexive, replacing b by a we get: a £ a Þ a = a is true.**(ii)** R is not symmetric, a £ b, but b £ a e.g. 2 < 3, but 3 < 2.**(iii)** R is transitive, if a £ b and b £ c, then a £ c, e.g. 2 < 3, 3 < 4 Þ 2 < 4.

**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.**

**Sol.** **(i)** (1, 1), (2, 2), (3, 3) do not belong to relation R. Þ R is not reflexive.**(ii)** It is symmetric since (1, 2) and (2, 1) belong to R.**(iii)** Q (1, 2) & (2, 1) Î R but (1, 1) R R not E is not 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.**

**Sol.** **(i)** The number of pages in a book remain the same. => R is reflexive.**(ii)** If the book x has the same number of pages as the book y.

=> book y has the same number of pages as the book x.

=> The relation R is symmetric.**(iii)** If books x and y have the same number of pages. Also books y and z have

the same number of pages.

=> Books x and z also have the same number of pages.

=> R is transitive. Thus, R 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} arerelated 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}.**

**Sol.** A = {1, 2, 3, 4, 5} and R = {(a, b) : |a – b| is even}

=> R = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (5, 1),(5, 3),(5,5)}.**(a) (i)** Reflexive: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) Î R. Þ R is reflexive.**(ii)** If |a – b| is even, then |b – a| is also even, e.g., (1, 3) Î R and (3, 1) Î R, etc.

=> R is symmetric.**(iii)** Further a – c = a – b + b – c

If |a – b| and |b – c| are even, then their sum |a – b + b – c| is also even.

=> |a – c| is even, e.g; (1, 3), (3, 5) Î R Þ (1, 5) Î R, etc

=> R is transitive. Hence R is an equivalence relation.

**(b)** Elements of {1, 3, 5} are related to each other.

Since |1 – 3| = 2, |3 – 5| = 2, |1 – 5| = 4. All are even numbers.

Similarly, elements of {2, 4} are related to each other. Since |2 – 4| = 2 an

even number.

No element of set {1, 3, 5} is related to any element of {2, 4}, since |1 – 2| = 1

not an even number.

**Question 9. Show that each of the relation R in the set A ={x not E 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 eachcase.**

**Sol. **The set A \ {x Î Z : 0 £ x £ 12} = {0, 1, 2, ………….., 12}**(i)** R = {(a, b) : |a – b| is a multiple of 4}

R = {(1, 5), (1, 9), (2, 6), (2, 10), (3, 7), (3, 11), (4, 8), (4, 12),

(5, 9), (6, 10), (7, 11), (8, 12), (5, 1), (9, 1), (6, 2), (10, 2), (7, 3),

(11, 3), (8, 4), (12, 4), (9, 5), (10, 6), (11, 7), (12, 8), (0, 0), (1, 1),

(2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), (10, 10), (11, 11), (12, 12)}

**(a)**(1, 1), (2, 2), ………..(12, 12) not E R. => R is reflexive.**(b)**(1, 5), (5, 1) Î R also (1, 9), (9, 1) Î R, etc. => R is symmetric.**(c)**(1, 5) & (5, 9) Î R Þ (1, 9) Î R, and so on. => R is transitive.

R is an equivalence relation.

The set related to 1 is {1, 5, 9}.

**(ii)** R = {(a, b) : a = b} = {(0, 0), (1, 1), (2, 2) ……. (12, 12)}

**(a)** a = a Þ (a, a) E R => R is reflexive.**(b)** Again if (a, b) ÎR, then (b, a) E R (: a = b).

=> R is symmetric.**(c)** If (a, b) Î R, (b, c) Î R and a = b & b = c Þ a = c Þ (a, c) ÎR

Hence, R is transitive.

The set related to 1 is {1}.

**Note:** If every element of a relation has the same image and preimage, then the relation is always equivalence i.e.; reflexive, symmetric as well as transitive.

**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**

**Sol.** Let A = set of straight lines in a plane**(i)** R : {(a, b) : a is perpendicular to b} let a, b be two perpendicular lines.**(a)** If line a is perpendicular to b then b is perpendicular to a

=> R is symmetric**(b)** Here a is not perpendicular to itself. => R is not reflexive.**(c)** If a is perpendicular to b and b is perpendicular to c, but a is not

perpendicular but will be parallel to c. => R is not transitive.

Thus R is symmetric but neither reflexive nor transitive.

**(ii)** Let A = set of real numbers R = {(a, b) : a > b}**(a)** An element is not greater than itself Þ R is not reflexive.**(b)** If a > b but b > a Þ R is not symmetric.

**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 1 (0, 0) is the circle passing through P with originas centre.**

**Sol.** Let O be the origin then the relation

R = {(P, Q) : OP = OQ}**(i)** Take any distance OP,

OP = PO Þ R is reflexive.**(ii)** R is symmetric, if OP = OQ then OQ = OP**(iii)** R is transitive, if OP = OQ and OQ = OR Þ OP = OR

Hence, R is an equivalence relation.

We know that collection of all points in a plane, which are at a fixed

distance from a fixed point in the plane, is called a circle.

Since OP = K (constant)

=> P is a variable point and lies on a circle with centre at the origin.

**Question 12. Show that the relation R defined in the set A of all triangles asR = {(T1, T2) : T1is 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, 10. Which triangles among T1, T2 and T3 are related?**

**Hint:** If the ratios of the corresponding sides of two triangles are equal, then the two triangles are similar.

**(b)** If triangle T1 is similar to triangle T2 then triangle T2 is similar to triangle T1 => R is symmetric.**(c)** Let triangle T1 is similar to triangle T2 and triangle T2 is similar to triangle T3 then triangle T1 is similar to triangle T3 => R is transitive. Hence, R is an equivalence relation.

**(ii)** Two triangles are similar if their sides are proportional now sides 3, 4, 5 of

triangle T1 are proportional to the sides 6, 8, 10 of triangle T3

: T1 is related to T3

**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?**

**Sol. **Let n be the number of sides of polygon P1.

R = {(P1, P2) : P1 and P2 are n sided polygons}

**(i) (a)** Any polygon P1 has n sides => R is reflexive

**(b)** If P1 has n sides, P2 also has n sides then if P2 has n sides P1 also has n sides. => R is symmetric.**(c)** Let P1, P2 are n sided polygons and P2 , P3 are n sided polygons.

Then, P1 and P3 are also n sided polygons. Þ R is transitive.

Hence R is an equivalence relation.**(ii)** We know that triangle is a polygon with 3 sides.

:. the set A = set of all the triangles in a plane.

**Question 14. Let L be the set of all lines in XY plane and R be the relation inL 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.**

**Note:** Two parallel lines have same slope.

**Sol.** L = set of all the lines in XY plane, R = {(L1, L2) : L1 is parallel to L2}

**(i) (a)** L1 is parallel to itself Þ R is reflexive.**(b)** L1 is parallel to L2

=> L2 is parallel to L1 => R is symmetric.

**(c)** Let L1 is parallel to L2 and L2 is parallel to L3 L1 is parallel to L3 => R is transitive. Hence, R is an equivalence relation.

**(ii)** y = 2x + 4 \ its slope = 2

We know that two parallel lines have same slope.

:. the set of lines parallel to y = 2x + 4 is:

y = 2x + k, where k is any arbitrary constant.

**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.**

### Exercise 1.2

**Question 1. Show that the function f : R -> R defined by f (x) =1/x is one-one 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?**

**Hint:** Consider f : A -> B, if every element of A has a unique image in B, then f is called one-one-function. If every element of B is an image of some element of A, then f is called onto function.

**Question 2. Check the injectivity and surjectivity of the following functions:**

**Hint:** The one-One function is also called injective function and onto function is also called the surjective function.

**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.**

**Sol**. f : R -> R given by f (x) = [x]**(a)** Here, f (1. 2) = 1 and f (1. 5) = 1 Þ f is not one-one**(b)** All the images of x Î R belonging to its domain have integers as the images in codomain. But no fraction proper or improper belonging to codomain of f has any pre-image in its domain.

E.g., there does not exist any element xÎR such that f(x) = 1.5.

=> f is not onto.

**Question 4. Show that the Modulus Function f : R -> R given by f (x) = |x|, is neither one-one 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 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**

**Hint:** If a function is one-one and onto, then the function is called bijective function.

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

**Question 9. Let f : N ->N be defined by**

**Question 10. Let A = R – {3} and B = R – {1}. consider the function f : A -> B defined by**

**Question 11. Let f : R -> R be defined as f (x) = x ^{4}. 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.**

**Sol.** **(d)** f (–1) = (–1)4 = 1 and f (1) = 14 = 1 =>–1, 1 have the same image 1 Þ f is not one-one

Further –2 in the codomain of f has no pre-image in its domain.

=> f is not onto Þ 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} andg : {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**

**Question 4. If f (x) = (4x 3)/(6x-4), x not = 2/3, show that fof (x) = x, for all not = 2/3 .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)}**

**Hint:** If f is one-one and onto, then f must be invertible.

**Question 6. Show that f : [– 1, 1] -> R, given by f (x) = x/(x+2) is one-one. Find the inverse of the function f : [– 1, 1] -> Range f.**

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

**Question 8.**

**Question 9. **

**Hint: **If f: X -> Y is a function such that there exists a function, g: Y X-> such that gof = IX and fog = IY , then f must be one-one and onto and hence f is invertible.

**Question 10. Let f : X -> Y be an invertible function. Show that f has unique inverse.**

**Hint: suppose g _{1 }and g_{2} are two inverses of f. Then for all y E Y, fog_{1}(y) = I_{y} = fog_{2} (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) = (3-x3 )1/3, then fof (x) is**

**Question 14. Let f : R -{– 4/3} ->R be a function defined as f (x) = 4x/3x +4. The inverse of f is the map g: Range f -> R- {– 4/3} given by**

### Exercise 1.4

**Note: **As per the latest syllabus provided by CBSE for the year 2019-20,

Topic – “Binary Operations” of chapter – “Relations and Functions” has been removed.

**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.**

**Question 2. For each binary operation * defined below, determine whether * is commutative or associative.**

**Question 3. 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.(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 the question no. 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**

**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:**

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

**Question 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 *on A, if any.**

**Question 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 Î N.(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a**

**Question 13. Consider a binary operation * on N defined as a * b = a ^{3}+b^{3}. 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?

### Miscellaneous Exercise

**Question 1. Let f : R –> R be defined as f (x) = 10x + 7. Find the function g : R –>R such that gof = fog = IR**

**Question 2. Let f: W ->W be defined as f (n) = n – 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.**

**Question 3. If f : R -> R is defined by f (x) = x ^{2} – 3x + 2, find f (f (x)).**

**Question 4. Show that the function f : R -> { x E R: – 1 < x < 1} defined by f (x) = x/(1+ |x|), x E R is one one and onto function.**

**Question 5. Show that the function f: R -> R given by f (x) = x ^{3} is injective.**

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

**Question 7. Give examples of two functions f : N -> N and g : N ->N such that gof is onto but f is not onto.**

**Question 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 C- B, Is R an equivalence rel ation on P (X)? Justify your answer.**

**Question 9. Given a non-empty set X, consider the binary operation * : P (X) × P (X) -> P(X) given by A * B = A Ç 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 onlyinvertible element in P (X) with respect to the operation *.**

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

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

**Question 12. Consider the binary operations * : R × R ® R and o : R × R ->R defined as a * b = |a – b| and a o b = a, ” a, b Î R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ” a, b, c E R, a * (b o c) = (a * b) o (a * b). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.**

**Question 13. Given a non-empty set X, let * : P(X) × P (X) ® P (X) be defined as****A * B = (A – B) È (B – A), ” A, B Î P (X). Show that the empty set f is theidentity for the operation * and all the elements A of P (X) are invertible withA**

^{–1}= A.

**Question 14. Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as**

**Question 15. Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A ® B be functions defined**

**Question 16. Let A = {1, 2, 3}. The number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is(a) 1 (b) 2 (c) 3 (d) 4**

**Question 17. Let A = {1, 2, 3}. The number of equivalence relations containing (1, 2) is (a) 1 (b) 2 (c) 3 (d) 4**

**Question 18. Let f : R ® R be the Signum function defined as **

**Question 19. Number of binary operations on the set {a, b} are(a) 10 (b) 16 (c) 20 (d) 8**

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