Mathematics · Textbook solutions

Relations and Functions

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 79 questions

1.2 Types of Relations

35 q

Solved Examples

Worked · 6
  1. 1.1 Eg.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}R = \{(a, b) : a \text{ is sister of } b\} is the empty relation and R′={(a,b):the difference between heights of a and b is less than 3 meters}R' = \{(a, b) : \text{the difference between heights of } a \text{ and } b \text{ is less than 3 meters}\} is the universal relation.
  2. 1.1 Eg.2
    Let T be the set of all triangles in a plane with R a relation in T given by R={(T1,T2):T1 is congruent to T2}R = \{(T_{1}, T_{2}) : T_{1} \text{ is congruent to } T_{2}\}. Show that R is an equivalence relation.
  3. 1.1 Eg.3
    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 = \{(L_{1}, L_{2}) : L_{1} \text{ is perpendicular to } L_{2}\}. Show that R is symmetric but neither reflexive nor transitive.
  4. 1.1 Eg.4
    Show that the relation R in the set {1,2,3}\{1, 2, 3\} given by R={(1,1),(2,2),(3,3),(1,2),(2,3)}R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\} is reflexive but neither symmetric nor transitive.
  5. 1.1 Eg.5
    Show that the relation R in the set Z\mathbf{Z} of integers given by R={(a,b):2 divides a−b}R = \{(a, b) : 2 \text{ divides } a - b\} is an equivalence relation.
  6. 1.1 Eg.6
    Let R be the relation defined in the set A={1,2,3,4,5,6,7}A = \{1, 2, 3, 4, 5, 6, 7\} by R={(a,b):both a and b are either odd or even}R = \{(a, b) : \text{both } a \text{ and } b \text{ are either odd or even}\}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7}\{1, 3, 5, 7\} are related to each other and all the elements of the subset {2,4,6}\{2, 4, 6\} are related to each other, but no element of the subset {1,3,5,7}\{1, 3, 5, 7\} is related to any element of the subset {2,4,6}\{2, 4, 6\}.

Exercise 1.1

Practice · 29
  1. Determine whether each of the following relations are reflexive, symmetric and transitive:
    Ex 1.1 Q1(i)
    Relation R in the set A={1,2,3,...,13,14}A = \{1, 2, 3, ..., 13, 14\} defined as R={(x,y):3x−y=0}R = \{(x, y) : 3x - y = 0\}
  2. Ex 1.1 Q1(ii)
    Relation R in the set N\mathbf{N} of natural numbers defined as R={(x,y):y=x+5 and x<4}R = \{(x, y) : y = x + 5 \text{ and } x < 4\}
  3. Ex 1.1 Q1(iii)
    Relation R in the set A={1,2,3,4,5,6}A = \{1, 2, 3, 4, 5, 6\} as R={(x,y):y is divisible by x}R = \{(x, y) : y \text{ is divisible by } x\}
  4. Ex 1.1 Q1(iv)
    Relation R in the set Z\mathbf{Z} of all integers defined as R={(x,y):x−y is an integer}R = \{(x, y) : x - y \text{ is an integer}\}
  5. Ex 1.1 Q1(v)(a)
    Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x and y work at the same place}R = \{(x, y) : x \text{ and } y \text{ work at the same place}\}
  6. Ex 1.1 Q1(v)(b)
    Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x and y live in the same locality}R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}
  7. Ex 1.1 Q1(v)(c)
    Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x is exactly 7 cm taller than y}R = \{(x, y) : x \text{ is exactly 7 cm taller than } y\}
  8. Ex 1.1 Q1(v)(d)
    Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x is wife of y}R = \{(x, y) : x \text{ is wife of } y\}
  9. Ex 1.1 Q1(v)(e)
    Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x is father of y}R = \{(x, y) : x \text{ is father of } y\}
  10. Ex 1.1 Q2
    Show that the relation R in the set R\mathbf{R} of real numbers, defined as R={(a,b):a≤b2}R = \{(a, b) : a \le b^{2}\} is neither reflexive nor symmetric nor transitive.
  11. Ex 1.1 Q3
    Check whether the relation R defined in the set {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} as R={(a,b):b=a+1}R = \{(a, b) : b = a + 1\} is reflexive, symmetric or transitive.
  12. Ex 1.1 Q4
    Show that the relation R in R\mathbf{R} defined as R={(a,b):a≤b}R = \{(a, b) : a \le b\}, is reflexive and transitive but not symmetric.
  13. Ex 1.1 Q5
    Check whether the relation R in R\mathbf{R} defined by R={(a,b):a≤b3}R = \{(a, b) : a \le b^{3}\} is reflexive, symmetric or transitive.
  14. Ex 1.1 Q6
    Show that the relation R in the set {1,2,3}\{1, 2, 3\} given by R={(1,2),(2,1)}R = \{(1, 2), (2, 1)\} is symmetric but neither reflexive nor transitive.
  15. Ex 1.1 Q7
    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}R = \{(x, y) : x \text{ and } y \text{ have same number of pages}\} is an equivalence relation.
  16. Ex 1.1 Q8
    Show that the relation R in the set A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} given by R={(a,b):∣a−b∣ is even}R = \{(a, b) : |a - b| \text{ is even}\}, is an equivalence relation. Show that all the elements of {1,3,5}\{1, 3, 5\} are related to each other and all the elements of {2,4}\{2, 4\} are related to each other. But no element of {1,3,5}\{1, 3, 5\} is related to any element of {2,4}\{2, 4\}.
  17. Show that each of the relation R in the set A={x∈Z:0≤x≤12}A = \{x \in \mathbf{Z} : 0 \le x \le 12\}, given by the following, is an equivalence relation. Find the set of all elements related to 1 in each case.
    Ex 1.1 Q9(i)
    R={(a,b):∣a−b∣ is a multiple of 4}R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}
  18. Ex 1.1 Q9(ii)
    R={(a,b):a=b}R = \{(a, b) : a = b\}
  19. Give an example of a relation. Which is
    Ex 1.1 Q10(i)
    Symmetric but neither reflexive nor transitive.
  20. Ex 1.1 Q10(ii)
    Transitive but neither reflexive nor symmetric.
  21. Ex 1.1 Q10(iii)
    Reflexive and symmetric but not transitive.
  22. Ex 1.1 Q10(iv)
    Reflexive and transitive but not symmetric.
  23. Ex 1.1 Q10(v)
    Symmetric and transitive but not reflexive.
  24. Ex 1.1 Q11
    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}R = \{(P, Q) : \text{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)P \ne (0, 0) is the circle passing through P with origin as centre.
  25. Ex 1.1 Q12
    Show that the relation R defined in the set A of all triangles as R={(T1,T2):T1 is similar to T2}R = \{(T_{1}, T_{2}) : T_{1} \text{ is similar to } T_{2}\}, is equivalence relation. Consider three right angle triangles T1T_{1} with sides 3, 4, 5, T2T_{2} with sides 5, 12, 13 and T3T_{3} with sides 6, 8, 10. Which triangles among T1T_{1}, T2T_{2} and T3T_{3} are related?
  26. Ex 1.1 Q13
    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}R = \{(P_{1}, P_{2}) : P_{1} \text{ and } P_{2} \text{ 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?
  27. Ex 1.1 Q14
    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}R = \{(L_{1}, L_{2}) : L_{1} \text{ is parallel to } L_{2}\}. Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4y = 2x + 4.
  28. Ex 1.1 Q15
    Let R be the relation in the set {1,2,3,4}\{1, 2, 3, 4\} given by R={(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}. Choose the correct answer.
    1. A.
      R is reflexive and symmetric but not transitive.
    2. B.
      R is reflexive and transitive but not symmetric.
    3. C.
      R is symmetric and transitive but not reflexive.
    4. D.
      R is an equivalence relation.
  29. Ex 1.1 Q16
    Let R be the relation in the set N\mathbf{N} given by R={(a,b):a=b−2, b>6}R = \{(a, b) : a = b - 2,\, b > 6\}. Choose the correct answer.
    1. A.
      (2,4)∈R(2, 4) \in R
    2. B.
      (3,8)∈R(3, 8) \in R
    3. C.
      (6,8)∈R(6, 8) \in R
    4. D.
      (8,7)∈R(8, 7) \in R

1.3 Types of Functions

25 q

Solved Examples

Worked · 8
  1. 1.2 Eg.7
    Let A be the set of all 50 students of Class X in a school. Let f:A→Nf : A \to \mathbf{N} be function defined by f(x)=f(x) = roll number of the student xx. Show that ff is one-one but not onto.
  2. 1.2 Eg.8
    Show that the function f:N→Nf : \mathbf{N} \to \mathbf{N}, given by f(x)=2xf(x) = 2x, is one-one but not onto.
  3. 1.2 Eg.9
    Prove that the function f:R→Rf : \mathbf{R} \to \mathbf{R}, given by f(x)=2xf(x) = 2x, is one-one and onto.
  4. 1.2 Eg.10
    Show that the function f:N→Nf : \mathbf{N} \to \mathbf{N}, given by f(1)=f(2)=1f(1) = f(2) = 1 and f(x)=x−1f(x) = x - 1, for every x>2x > 2, is onto but not one-one.
  5. 1.2 Eg.11
    Show that the function f:R→Rf : \mathbf{R} \to \mathbf{R}, defined as f(x)=x2f(x) = x^{2}, is neither one-one nor onto.
  6. 1.2 Eg.12
    Show that f:N→Nf : \mathbf{N} \to \mathbf{N}, given by f(x)={x+1,if x is oddx−1,if x is evenf(x) = \begin{cases} x + 1, & \text{if } x \text{ is odd} \\ x - 1, & \text{if } x \text{ is even} \end{cases} is both one-one and onto.
  7. 1.2 Eg.13
    Show that an onto function f:{1,2,3}→{1,2,3}f : \{1, 2, 3\} \to \{1, 2, 3\} is always one-one.
  8. 1.2 Eg.14
    Show that a one-one function f:{1,2,3}→{1,2,3}f : \{1, 2, 3\} \to \{1, 2, 3\} must be onto.

Exercise 1.2

Practice · 17
  1. Ex 1.2 Q1
    Show that the function f:R∗→R∗f : \mathbf{R}_{*} \to \mathbf{R}_{*} defined by f(x)=1xf(x) = \frac{1}{x} is one-one and onto, where R∗\mathbf{R}_{*} is the set of all non-zero real numbers. Is the result true, if the domain R∗\mathbf{R}_{*} is replaced by N\mathbf{N} with co-domain being same as R∗\mathbf{R}_{*}?
  2. Check the injectivity and surjectivity of the following functions:
    Ex 1.2 Q2(i)
    f:N→Nf : \mathbf{N} \to \mathbf{N} given by f(x)=x2f(x) = x^{2}
  3. Ex 1.2 Q2(ii)
    f:Z→Zf : \mathbf{Z} \to \mathbf{Z} given by f(x)=x2f(x) = x^{2}
  4. Ex 1.2 Q2(iii)
    f:R→Rf : \mathbf{R} \to \mathbf{R} given by f(x)=x2f(x) = x^{2}
  5. Ex 1.2 Q2(iv)
    f:N→Nf : \mathbf{N} \to \mathbf{N} given by f(x)=x3f(x) = x^{3}
  6. Ex 1.2 Q2(v)
    f:Z→Zf : \mathbf{Z} \to \mathbf{Z} given by f(x)=x3f(x) = x^{3}
  7. Ex 1.2 Q3
    Prove that the Greatest Integer Function f:R→Rf : \mathbf{R} \to \mathbf{R}, given by f(x)=[x]f(x) = [x], is neither one-one nor onto, where [x][x] denotes the greatest integer less than or equal to xx.
  8. Ex 1.2 Q4
    Show that the Modulus Function f:R→Rf : \mathbf{R} \to \mathbf{R}, given by f(x)=∣x∣f(x) = |x|, is neither one-one nor onto, where ∣x∣|x| is xx, if xx is positive or 0 and ∣x∣|x| is −x-x, if xx is negative.
  9. Ex 1.2 Q5
    Show that the Signum Function f:R→Rf : \mathbf{R} \to \mathbf{R}, given by f(x)={1,if x>00,if x=01,if x<0f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ 1, & \text{if } x < 0 \end{cases} is neither one-one nor onto.
  10. Ex 1.2 Q6
    Let A={1,2,3}A = \{1, 2, 3\}, B={4,5,6,7}B = \{4, 5, 6, 7\} and let f={(1,4),(2,5),(3,6)}f = \{(1, 4), (2, 5), (3, 6)\} be a function from A to B. Show that ff is one-one.
  11. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
    Ex 1.2 Q7(i)
    f:R→Rf : \mathbf{R} \to \mathbf{R} defined by f(x)=3−4xf(x) = 3 - 4x
  12. Ex 1.2 Q7(ii)
    f:R→Rf : \mathbf{R} \to \mathbf{R} defined by f(x)=1+x2f(x) = 1 + x^{2}
  13. Ex 1.2 Q8
    Let A and B be sets. Show that f:A×B→B×Af : A \times B \to B \times A such that f(a,b)=(b,a)f(a, b) = (b, a) is bijective function.
  14. Ex 1.2 Q9
    Let f:N→Nf : \mathbf{N} \to \mathbf{N} be defined by f(n)={n+12,if n is oddn2,if n is evenf(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} for all n∈Nn \in \mathbf{N}. State whether the function ff is bijective. Justify your answer.
  15. Ex 1.2 Q10
    Let A=R−{3}A = \mathbf{R} - \{3\} and B=R−{1}B = \mathbf{R} - \{1\}. Consider the function f:A→Bf : A \to B defined by f(x)=(x−2x−3)f(x) = \left(\frac{x - 2}{x - 3}\right). Is ff one-one and onto? Justify your answer.
  16. Ex 1.2 Q11
    Let f:R→Rf : \mathbf{R} \to \mathbf{R} be defined as f(x)=x4f(x) = x^{4}. Choose the correct answer.
    1. A.
      ff is one-one onto
    2. B.
      ff is many-one onto
    3. C.
      ff is one-one but not onto
    4. D.
      ff is neither one-one nor onto.
  17. Ex 1.2 Q12
    Let f:R→Rf : \mathbf{R} \to \mathbf{R} be defined as f(x)=3xf(x) = 3x. Choose the correct answer.
    1. A.
      ff is one-one onto
    2. B.
      ff is many-one onto
    3. C.
      ff is one-one but not onto
    4. D.
      ff is neither one-one nor onto.

Miscellaneous Exercise on Chapter 1

19 q

Solved Examples

Worked · 12
  1. Misc Eg.15
    Let f:{2,3,4,5}→{3,4,5,9}f : \{2, 3, 4, 5\} \to \{3, 4, 5, 9\} and g:{3,4,5,9}→{7,11,15}g : \{3, 4, 5, 9\} \to \{7, 11, 15\} be functions defined as f(2)=3f(2) = 3, f(3)=4f(3) = 4, f(4)=f(5)=5f(4) = f(5) = 5 and g(3)=g(4)=7g(3) = g(4) = 7 and g(5)=g(9)=11g(5) = g(9) = 11. Find gofgof.
  2. Misc Eg.16
    Find gofgof and fogfog, if f:R→Rf : \mathbf{R} \to \mathbf{R} and g:R→Rg : \mathbf{R} \to \mathbf{R} are given by f(x)=cos⁡xf(x) = \cos x and g(x)=3x2g(x) = 3x^2. Show that gof≠foggof \ne fog.
  3. Misc Eg.17
    Let f:N→Yf : \mathbf{N} \to \mathrm{Y} be a function defined as f(x)=4x+3f(x) = 4x + 3, where, Y={y∈N:y=4x+3 for some x∈N}\mathrm{Y} = \{y \in \mathbf{N} : y = 4x + 3 \text{ for some } x \in \mathbf{N}\}. Show that ff is invertible. Find the inverse.
  4. Misc Eg.18
    If R1\mathrm{R}_1 and R2\mathrm{R}_2 are equivalence relations in a set A, show that R1∩R2\mathrm{R}_1 \cap \mathrm{R}_2 is also an equivalence relation.
  5. Misc Eg.19
    Let R be a relation on the set A of ordered pairs of positive integers defined by (x,y) R (u,v)(x, y)\ \mathrm{R}\ (u, v) if and only if xv=yuxv = yu. Show that R is an equivalence relation.
  6. Misc Eg.20
    Let X={1,2,3,4,5,6,7,8,9}\mathrm{X} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}. Let R1\mathrm{R}_1 be a relation in X given by R1={(x,y):x−y is divisible by 3}\mathrm{R}_1 = \{(x, y) : x - y \text{ is divisible by } 3\} and R2\mathrm{R}_2 be another relation on X given by R2={(x,y):{x,y}⊂{1,4,7}}\mathrm{R}_2 = \{(x, y): \{x, y\} \subset \{1, 4, 7\}\} or {x,y}⊂{2,5,8}\{x, y\} \subset \{2, 5, 8\} or {x,y}⊂{3,6,9}}\{x, y\} \subset \{3, 6, 9\}\}. Show that R1=R2\mathrm{R}_1 = \mathrm{R}_2.
  7. Misc Eg.21
    Let f:X→Yf : \mathrm{X} \to \mathrm{Y} be a function. Define a relation R in X given by R={(a,b):f(a)=f(b)}\mathrm{R} = \{(a, b): f(a) = f(b)\}. Examine whether R is an equivalence relation or not.
  8. Misc Eg.22
    Find the number of all one-one functions from set A={1,2,3}\mathrm{A} = \{1, 2, 3\} to itself.
  9. Misc Eg.23
    Let A={1,2,3}\mathrm{A} = \{1, 2, 3\}. Then show that the number of relations containing (1,2)(1, 2) and (2,3)(2, 3) which are reflexive and transitive but not symmetric is three.
  10. Misc Eg.24
    Show that the number of equivalence relation in the set {1,2,3}\{1, 2, 3\} containing (1,2)(1, 2) and (2,1)(2, 1) is two.
  11. Misc Eg.25
    Consider the identity function IN:N→N\mathrm{I}_{\mathbf{N}} : \mathbf{N} \to \mathbf{N} defined as IN(x)=x ∀ x∈N\mathrm{I}_{\mathbf{N}}(x) = x\ \forall\ x \in \mathbf{N}. Show that although IN\mathrm{I}_{\mathbf{N}} is onto but IN+IN:N→N\mathrm{I}_{\mathbf{N}} + \mathrm{I}_{\mathbf{N}} : \mathbf{N} \to \mathbf{N} defined as (IN+IN)(x)=IN(x)+IN(x)=x+x=2x(\mathrm{I}_{\mathbf{N}} + \mathrm{I}_{\mathbf{N}})(x) = \mathrm{I}_{\mathbf{N}}(x) + \mathrm{I}_{\mathbf{N}}(x) = x + x = 2x is not onto.
  12. Misc Eg.26
    Consider a function f:[0,π2]→Rf : \left[0, \frac{\pi}{2}\right] \to \mathbf{R} given by f(x)=sin⁡xf(x) = \sin x and g:[0,π2]→Rg : \left[0, \frac{\pi}{2}\right] \to \mathbf{R} given by g(x)=cos⁡xg(x) = \cos x. Show that ff and gg are one-one, but f+gf + g is not one-one.

Miscellaneous Exercise

Practice · 7
  1. Misc Q1
    Show that the function f:R→{x∈R:−1<x<1}f : \mathbf{R} \to \{x \in \mathbf{R} : -1 < x < 1\} defined by f(x)=x1+∣x∣f(x) = \frac{x}{1 + |x|}, x∈Rx \in \mathbf{R} is one one and onto function.
  2. Misc Q2
    Show that the function f:R→Rf : \mathbf{R} \to \mathbf{R} given by f(x)=x3f(x) = x^3 is injective.
  3. Misc Q3
    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\mathrm{A} \subset \mathrm{B}. Is R an equivalence relation on P(X)? Justify your answer.
  4. Misc Q4
    Find the number of all onto functions from the set {1,2,3,…,n}\{1, 2, 3, \ldots, n\} to itself.
  5. Misc Q5
    Let A={−1,0,1,2}\mathrm{A} = \{-1, 0, 1, 2\}, B={−4,−2,0,2}\mathrm{B} = \{-4, -2, 0, 2\} and f,g:A→Bf, g : \mathrm{A} \to \mathrm{B} be functions defined by f(x)=x2−xf(x) = x^2 - x, x∈Ax \in \mathrm{A} and g(x)=2∣x−12∣−1g(x) = 2\left|x - \frac{1}{2}\right| - 1, x∈Ax \in \mathrm{A}. Are ff and gg equal? Justify your answer. (Hint: One may note that two functions f:A→Bf : \mathrm{A} \to \mathrm{B} and g:A→Bg : \mathrm{A} \to \mathrm{B} such that f(a)=g(a) ∀ a∈Af(a) = g(a)\ \forall\ a \in \mathrm{A}, are called equal functions).
  6. Misc Q6
    Let A={1,2,3}\mathrm{A} = \{1, 2, 3\}. Then number of relations containing (1,2)(1, 2) and (1,3)(1, 3) which are reflexive and symmetric but not transitive is
    1. A.
      1
    2. B.
      2
    3. C.
      3
    4. D.
      4
  7. Misc Q7
    Let A={1,2,3}\mathrm{A} = \{1, 2, 3\}. Then number of equivalence relations containing (1,2)(1, 2) is
    1. A.
      1
    2. B.
      2
    3. C.
      3
    4. D.
      4