Mathematics · Textbook solutions

Relations and Functions

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 76 questions

2.2 Cartesian Products of Sets

19 q

Solved Examples

Worked · 6
  1. 2.1 Eg.1
    If (x+1,y−2)=(3,1)(x + 1, y - 2) = (3,1), find the values of xx and yy.
  2. 2.1 Eg.2
    If P={a,b,c}P = \{a, b, c\} and Q={r}Q = \{r\}, form the sets P×QP \times Q and Q×PQ \times P. Are these two products equal?
  3. 2.1 Eg.3
    Let A={1,2,3}A = \{1,2,3\}, B={3,4}B = \{3,4\} and C={4,5,6}C = \{4,5,6\}. Find (i) A×(B∩C)A \times (B \cap C) (ii) (A×B)∩(A×C)(A \times B) \cap (A \times C) (iii) A×(B∪C)A \times (B \cup C) (iv) (A×B)∪(A×C)(A \times B) \cup (A \times C)
  4. 2.1 Eg.4
    If P={1,2}P = \{1, 2\}, form the set P×P×PP \times P \times P.
  5. 2.1 Eg.5
    If R\mathbb{R} is the set of all real numbers, what do the cartesian products R×R\mathbb{R} \times \mathbb{R} and R×R×R\mathbb{R} \times \mathbb{R} \times \mathbb{R} represent?
  6. 2.1 Eg.6
    If A×B={(p,q),(p,r),(m,q),(m,r)}A \times B = \{(p, q), (p, r), (m, q), (m, r)\}, find AA and BB.

Exercise 2.1

Practice · 13
  1. Ex 2.1 Q1
    If (x3+1,y−23)=(53,13)\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right), find the values of xx and yy.
  2. Ex 2.1 Q2
    If the set AA has 3 elements and the set B={3,4,5}B = \{3, 4, 5\}, then find the number of elements in (A×B)(A \times B).
  3. Ex 2.1 Q3
    If G={7,8}G = \{7, 8\} and H={5,4,2}H = \{5, 4, 2\}, find G×HG \times H and H×GH \times G.
  4. State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
    Ex 2.1 Q4(i)
    If P={m,n}P = \{m, n\} and Q={n,m}Q = \{n, m\}, then P×Q={(m,n),(n,m)}P \times Q = \{(m, n), (n, m)\}.
  5. Ex 2.1 Q4(ii)
    If AA and BB are non-empty sets, then A×BA \times B is a non-empty set of ordered pairs (x,y)(x, y) such that x∈Ax \in A and y∈By \in B.
  6. Ex 2.1 Q4(iii)
    If A={1,2}A = \{1, 2\}, B={3,4}B = \{3, 4\}, then A×(B∩ϕ)=ϕA \times (B \cap \phi) = \phi.
  7. Ex 2.1 Q5
    If A={−1,1}A = \{-1, 1\}, find A×A×AA \times A \times A.
  8. Ex 2.1 Q6
    If A×B={(a,x),(a,y),(b,x),(b,y)}A \times B = \{(a, x), (a, y), (b, x), (b, y)\}. Find AA and BB.
  9. Let A={1,2}A = \{1, 2\}, B={1,2,3,4}B = \{1, 2, 3, 4\}, C={5,6}C = \{5, 6\} and D={5,6,7,8}D = \{5, 6, 7, 8\}. Verify that
    Ex 2.1 Q7(i)
    A×(B∩C)=(A×B)∩(A×C)A \times (B \cap C) = (A \times B) \cap (A \times C).
  10. Ex 2.1 Q7(ii)
    A×CA \times C is a subset of B×DB \times D.
  11. Ex 2.1 Q8
    Let A={1,2}A = \{1, 2\} and B={3,4}B = \{3, 4\}. Write A×BA \times B. How many subsets will A×BA \times B have? List them.
  12. Ex 2.1 Q9
    Let AA and BB be two sets such that n(A)=3n(A) = 3 and n(B)=2n(B) = 2. If (x,1)(x, 1), (y,2)(y, 2), (z,1)(z, 1) are in A×BA \times B, find AA and BB, where xx, yy and zz are distinct elements.
  13. Ex 2.1 Q10
    The Cartesian product A×AA \times A has 9 elements among which are found (−1,0)(-1, 0) and (0,1)(0,1). Find the set AA and the remaining elements of A×AA \times A.

2.3 Relations

14 q

Solved Examples

Worked · 3
  1. 2.2 Eg.7
    Let A={1,2,3,4,5,6}A = \{1, 2, 3, 4, 5, 6\}. Define a relation RR from AA to AA by R={(x,y):y=x+1}R = \{(x, y) : y = x + 1\} (i) Depict this relation using an arrow diagram. (ii) Write down the domain, codomain and range of RR.
  2. 2.2 Eg.8
    The Fig 2.6 shows a relation between the sets PP and QQ. Write this relation (i) in set-builder form, (ii) in roster form. What is its domain and range?
  3. 2.2 Eg.9
    Let A={1,2}A = \{1, 2\} and B={3,4}B = \{3, 4\}. Find the number of relations from AA to BB.

Exercise 2.2

Practice · 11
  1. Ex 2.2 Q1
    Let A={1,2,3,…,14}A = \{1, 2, 3,\ldots,14\}. Define a relation RR from AA to AA by R={(x,y):3x−y=0R = \{(x, y) : 3x - y = 0, where x,y∈A}x, y \in A\}. Write down its domain, codomain and range.
  2. Ex 2.2 Q2
    Define a relation RR on the set N\mathbb{N} of natural numbers by R={(x,y):y=x+5R = \{(x, y) : y = x + 5, xx is a natural number less than 4; x,y∈N}x, y \in \mathbb{N}\}. Depict this relationship using roster form. Write down the domain and the range.
  3. Ex 2.2 Q3
    A={1,2,3,5}A = \{1, 2, 3, 5\} and B={4,6,9}B = \{4, 6, 9\}. Define a relation RR from AA to BB by R={(x,y)R = \{(x, y): the difference between xx and yy is odd; x∈Ax \in A, y∈B}y \in B\}. Write RR in roster form.
  4. Ex 2.2 Q4
    The Fig 2.7 shows a relationship between the sets PP and QQ. Write this relation (i) in set-builder form (ii) roster form. What is its domain and range?
  5. Let A={1,2,3,4,6}A = \{1, 2, 3, 4, 6\}. Let RR be the relation on AA defined by {(a,b):a,b∈A\{(a, b): a, b \in A, bb is exactly divisible by a}a\}.
    Ex 2.2 Q5(i)
    Write RR in roster form.
  6. Ex 2.2 Q5(ii)
    Find the domain of RR.
  7. Ex 2.2 Q5(iii)
    Find the range of RR.
  8. Ex 2.2 Q6
    Determine the domain and range of the relation RR defined by R={(x,x+5):x∈{0,1,2,3,4,5}}R = \{(x, x + 5) : x \in \{0, 1, 2, 3, 4, 5\}\}.
  9. Ex 2.2 Q7
    Write the relation R={(x,x3):xR = \{(x, x^{3}) : x is a prime number less than 10}\} in roster form.
  10. Ex 2.2 Q8
    Let A={x,y,z}A = \{x, y, z\} and B={1,2}B = \{1, 2\}. Find the number of relations from AA to BB.
  11. Ex 2.2 Q9
    Let RR be the relation on Z\mathbb{Z} defined by R={(a,b):a,b∈ZR = \{(a,b): a, b \in \mathbb{Z}, a−ba - b is an integer}\}. Find the domain and range of RR.

2.4 Functions

23 q

Solved Examples

Worked · 8
  1. 2.3 Eg.10
    Let N\mathbb{N} be the set of natural numbers and the relation RR be defined on N\mathbb{N} such that R={(x,y):y=2x,x,y∈N}R = \{(x, y) : y = 2x, x, y \in \mathbb{N}\}. What is the domain, codomain and range of RR? Is this relation a function?
  2. 2.3 Eg.11
    Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not? (i) R={(2,1),(3,1),(4,2)}R = \{(2,1), (3,1), (4,2)\}, (ii) R={(2,2),(2,4),(3,3),(4,4)}R = \{(2,2), (2,4), (3,3), (4,4)\} (iii) R={(1,2),(2,3),(3,4),(4,5),(5,6),(6,7)}R = \{(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)\}
  3. 2.3 Eg.12
    Let N\mathbb{N} be the set of natural numbers. Define a real valued function f:N→Nf : \mathbb{N} \to \mathbb{N} by f(x)=2x+1f(x) = 2x + 1. Using this definition, complete the table given below.
    xx1234567
    yyf(1)=…f(1) = \ldotsf(2)=…f(2) = \ldotsf(3)=…f(3) = \ldotsf(4)=…f(4) = \ldotsf(5)=…f(5) = \ldotsf(6)=…f(6) = \ldotsf(7)=…f(7) = \ldots
  4. 2.3 Eg.13
    Define the function f:R→Rf: \mathbb{R} \to \mathbb{R} by y=f(x)=x2y = f(x) = x^{2}, x∈Rx \in \mathbb{R}. Complete the Table given below by using this definition. What is the domain and range of this function? Draw the graph of ff.
    xx−4-4−3-3−2-2−1-101234
    y=f(x)=x2y = f(x) = x^{2}
  5. 2.3 Eg.14
    Draw the graph of the function f:R→Rf : \mathbb{R} \to \mathbb{R} defined by f(x)=x3f(x) = x^{3}, x∈Rx \in \mathbb{R}.
  6. 2.3 Eg.15
    Define the real valued function f:R−{0}→Rf : \mathbb{R} - \{0\} \to \mathbb{R} defined by f(x)=1xf(x) = \frac{1}{x}, x∈R−{0}x \in \mathbb{R} - \{0\}. Complete the Table given below using this definition. What is the domain and range of this function?
    xx−2-2−1.5-1.5−1-1−0.5-0.50.250.511.52
    y=1xy = \frac{1}{x}…\ldots…\ldots…\ldots…\ldots…\ldots…\ldots…\ldots…\ldots…\ldots
  7. 2.3 Eg.16
    Let f(x)=x2f(x) = x^{2} and g(x)=2x+1g(x) = 2x + 1 be two real functions. Find (f+g)(x)(f + g)(x), (f−g)(x)(f - g)(x), (fg)(x)(fg)(x), (fg)(x)\left(\frac{f}{g}\right)(x).
  8. 2.3 Eg.17
    Let f(x)=xf(x) = \sqrt{x} and g(x)=xg(x) = x be two functions defined over the set of non-negative real numbers. Find (f+g)(x)(f + g)(x), (f−g)(x)(f - g)(x), (fg)(x)(fg)(x) and (fg)(x)\left(\frac{f}{g}\right)(x).

Exercise 2.3

Practice · 15
  1. Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
    Ex 2.3 Q1(i)
    {(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)}\{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)\}
  2. Ex 2.3 Q1(ii)
    {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}\{(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)\}
  3. Ex 2.3 Q1(iii)
    {(1,3),(1,5),(2,5)}\{(1,3), (1,5), (2,5)\}.
  4. Find the domain and range of the following real functions:
    Ex 2.3 Q2(i)
    f(x)=−∣x∣f(x) = -|x|
  5. Ex 2.3 Q2(ii)
    f(x)=9−x2f(x) = \sqrt{9 - x^{2}}.
  6. A function ff is defined by f(x)=2x−5f(x) = 2x - 5. Write down the values of
    Ex 2.3 Q3(i)
    f(0)f(0)
  7. Ex 2.3 Q3(ii)
    f(7)f(7)
  8. Ex 2.3 Q3(iii)
    f(−3)f(-3).
  9. The function 'tt' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C)=9C5+32t(C) = \frac{9C}{5} + 32. Find
    Ex 2.3 Q4(i)
    t(0)t(0)
  10. Ex 2.3 Q4(ii)
    t(28)t(28)
  11. Ex 2.3 Q4(iii)
    t(−10)t(-10)
  12. Ex 2.3 Q4(iv)
    The value of CC, when t(C)=212t(C) = 212.
  13. Find the range of each of the following functions.
    Ex 2.3 Q5(i)
    f(x)=2−3xf(x) = 2 - 3x, x∈Rx \in \mathbb{R}, x>0x > 0.
  14. Ex 2.3 Q5(ii)
    f(x)=x2+2f(x) = x^{2} + 2, xx is a real number.
  15. Ex 2.3 Q5(iii)
    f(x)=xf(x) = x, xx is a real number.

Miscellaneous Exercise on Chapter 2

20 q

Miscellaneous Examples

Worked · 5
  1. Misc Eg.18
    Let R\mathbb{R} be the set of real numbers. Define the real function f:R→Rf : \mathbb{R} \to \mathbb{R} by f(x)=x+10f(x) = x + 10 and sketch the graph of this function.
  2. Misc Eg.19
    Let RR be a relation from Q\mathbb{Q} to Q\mathbb{Q} defined by R={(a,b):a,b∈QR = \{(a,b): a, b \in \mathbb{Q} and a−b∈Z}a - b \in \mathbb{Z}\}. Show that (i) (a,a)∈R(a,a) \in R for all a∈Qa \in \mathbb{Q} (ii) (a,b)∈R(a,b) \in R implies that (b,a)∈R(b, a) \in R (iii) (a,b)∈R(a,b) \in R and (b,c)∈R(b,c) \in R implies that (a,c)∈R(a,c) \in R
  3. Misc Eg.20
    Let f={(1,1),(2,3),(0,−1),(−1,−3)}f = \{(1,1), (2,3), (0, -1), (-1, -3)\} be a linear function from Z\mathbb{Z} into Z\mathbb{Z}. Find f(x)f(x).
  4. Misc Eg.21
    Find the domain of the function f(x)=x2+3x+5x2−5x+4f(x) = \frac{x^{2} + 3x + 5}{x^{2} - 5x + 4}
  5. Misc Eg.22
    The function ff is defined by f(x)={1−x,x<01,x=0x+1,x>0f(x) = \begin{cases} 1 - x, & x < 0 \\ 1, & x = 0 \\ x + 1, & x > 0 \end{cases} Draw the graph of f(x)f(x).

Miscellaneous Exercise

Practice · 15
  1. Misc Q1
    The relation ff is defined by f(x)={x2,0≤x≤33x,3≤x≤10f(x) = \begin{cases} x^{2}, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases} The relation gg is defined by g(x)={x2,0≤x≤23x,2≤x≤10g(x) = \begin{cases} x^{2}, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases} Show that ff is a function and gg is not a function.
  2. Misc Q2
    If f(x)=x2f(x) = x^{2}, find f(1.1)−f(1)(1.1−1)\frac{f(1.1) - f(1)}{(1.1 - 1)}.
  3. Misc Q3
    Find the domain of the function f(x)=x2+2x+1x2−8x+12f(x) = \frac{x^{2} + 2x + 1}{x^{2} - 8x + 12}.
  4. Misc Q4
    Find the domain and the range of the real function ff defined by f(x)=(x−1)f(x) = \sqrt{(x - 1)}.
  5. Misc Q5
    Find the domain and the range of the real function ff defined by f(x)=∣x−1∣f(x) = |x - 1|.
  6. Misc Q6
    Let f={(x,x21+x2):x∈R}f = \left\{\left(x, \frac{x^{2}}{1 + x^{2}}\right) : x \in \mathbb{R}\right\} be a function from R\mathbb{R} into R\mathbb{R}. Determine the range of ff.
  7. Misc Q7
    Let f,g:R→Rf, g : \mathbb{R} \to \mathbb{R} be defined, respectively by f(x)=x+1f(x) = x + 1, g(x)=2x−3g(x) = 2x - 3. Find f+gf + g, f−gf - g and fg\frac{f}{g}.
  8. Misc Q8
    Let f={(1,1),(2,3),(0,−1),(−1,−3)}f = \{(1,1), (2,3), (0,-1), (-1, -3)\} be a function from Z\mathbb{Z} to Z\mathbb{Z} defined by f(x)=ax+bf(x) = ax + b, for some integers aa, bb. Determine aa, bb.
  9. Let RR be a relation from N\mathbb{N} to N\mathbb{N} defined by R={(a,b):a,b∈NR = \{(a, b) : a, b \in \mathbb{N} and a=b2}a = b^{2}\}. Are the following true? Justify your answer in each case.
    Misc Q9(i)
    (a,a)∈R(a,a) \in R, for all a∈Na \in \mathbb{N}
  10. Misc Q9(ii)
    (a,b)∈R(a,b) \in R, implies (b,a)∈R(b,a) \in R
  11. Misc Q9(iii)
    (a,b)∈R(a,b) \in R, (b,c)∈R(b,c) \in R implies (a,c)∈R(a,c) \in R.
  12. Let A={1,2,3,4}A = \{1,2,3,4\}, B={1,5,9,11,15,16}B = \{1,5,9,11,15,16\} and f={(1,5),(2,9),(3,1),(4,5),(2,11)}f = \{(1,5), (2,9), (3,1), (4,5), (2,11)\}. Are the following true? Justify your answer in each case.
    Misc Q10(i)
    ff is a relation from AA to BB
  13. Misc Q10(ii)
    ff is a function from AA to BB.
  14. Misc Q11
    Let ff be the subset of Z×Z\mathbb{Z} \times \mathbb{Z} defined by f={(ab,a+b):a,b∈Z}f = \{(ab, a + b) : a, b \in \mathbb{Z}\}. Is ff a function from Z\mathbb{Z} to Z\mathbb{Z}? Justify your answer.
  15. Misc Q12
    Let A={9,10,11,12,13}A = \{9,10,11,12,13\} and let f:A→Nf : A \to \mathbb{N} be defined by f(n)f(n) = the highest prime factor of nn. Find the range of ff.