Relations and Functions
Exercise 1.1
1. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A={1,2,3,…,13,14} defined as R={(x,y) : 3x–y=0}
(ii) Relation R in the set N of natural numbers defined as R={(x,y) : y=x+5 and x<4}
(iii) Relation R in the set A={1,2,3,4,5,6} as R={(x,y) : y is divisible by x}
(iv) Relation R in the set Z of all integers defined as R={(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}
9. Show that each of the relation R in the set A={x\(\in\)Z : 0≤x≤12}, given by
(i) R={(a,b) : |a–b| is a multiple of 4}
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.
(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}. Choose the correct answer.
Relations and Functions
Exercise 1.2
1. Show that the function f: R∗→R∗ defined by \(f(x)=\frac{1}{x}\) is one-one and 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∗?
2. Check the injectivity and surjectivity of the following functions:
(i) f: N→N given by \(f(x)=x^2\).
(ii) f: Z→Z given by \(f(x)=x^2\).
(iii) f: R→R given by \(f(x)=x^2\).
(iv) f: N→N given by \(f(x)=x^3\).
(v) f: Z→Z given by \(f(x)=x^3\).
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.
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.
5. Show that the Signum Function f: R→R, given by:
\(f(x)=\begin{cases}1, \text{if}\:\: x>0\\0, \text{if}\:\: x=0\\1, \text{if}\:\: x<0\end{cases}\) is neither one-one nor onto.
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.
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
8. Let A and B be sets. Show that f: A×B→B×A such that f(a,b)=(b,a) is bijective function.
9. Let f: N→N be defined by
\(f(n)=\begin{cases}\frac{n+1}{2}, \text{if n is odd}\\\frac{n}{2}, \text{if n is even}\end{cases}\) for all \(n\in N\).
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 by \(f(x)=(\frac{x-2}{x-3})\). 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.
Relations and Functions
Miscellaneous Exercise
1. Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by \(f(x)={x\over {1+|x|}}\), x∈R is one one and onto function.
2. Show that the function f : R → R given by \(f(x) = x^3\) is injective.
3. 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 relation on P(X)? Justify your answer.
4. Find the number of all onto functions from the set {1, 2, 3, … , n} to itself.
5. Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined by \(f (x) = x^2 – x\), x ∈ A and \(g(x)=2|x-{1\over 2}|-1\),x∈A. Are f and g equal? Justify your answer.
6. Let A = {1, 2, 3}. Then 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
7. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:
(A) 1 (B) 2 (C) 3 (D) 4