Page No 5:
Question 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 as 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}
Page No 5:
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.
Page No 5:
Question 3:
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Page No 5:
Question 4:
Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Page No 5:
Question 5:
Check whether the relation R in R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric, or transitive.
Page No 6:
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 the same number of pages} is an equivalence relation.
Page No 6:
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 point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
Page No 6:
Question 12:
Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is 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?
Page No 6:
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?
Page No 6:
Question 14:
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}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
Page No 7:
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.
(A) (2, 4) ∈ R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 7) ∈ R
Page No 10:
Question 1:
Show that the function f: R* → R* defined by
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*?
Page No 10:
Question 2:
Check the injectivity and surjectivity of the following functions:
(i) f: N → N given by f(x) = x2
(ii) f: Z → Z given by f(x) = x2
(iii) f: R → R given by f(x) = x2
(iv) f: N → N given by f(x) = x3
(v) f: Z → Z given by f(x) = x3
Page No 10:
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.
Page No 11:
Question 4:
Show that the Modulus Function f: R → R given by
, is neither one-one nor onto, where
is x, if x is positive or 0 andis − x, if x is negative.
Page No 11:
Question 5:
Show that the Signum Function f: R → R, given by
is neither one-one nor onto.
Page No 11:
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
Page No 11:
Question 8:
Let A and B be sets. Show that f: A × B → B × A such that (a, b) = (b, a) is bijective function.
Page No 11:
Question 9:
Let f: N → N be defined by
State whether the function f is bijective. Justify your answer.
Page No 11:
Question 10:
Let A = R − {3} and B = R − {1}. Consider the function f: A → B defined by
. Is f one-one and onto? Justify your answer.
Page No 11:
Question 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
Page No 11:
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
Page No 18:
Question 1:
Let f: {1, 3, 4} → {1, 2, 5} and g: {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.
Page No 18:
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)}
is one-one. Find the inverse of the function f: [−1, 1] → Range f.
, for some x in [−1, 1], i.e.,
)Page No 18:
Question 7:
Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Page No 18:
Question 8:
Consider f: R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given f by
, where R+ is the set of all non-negative real numbers.
Page No 19:
Question 9:
Consider f: R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with
Page No 19:
Question 10:
Let f: X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y,
fog1(y) = IY(y) = fog2(y). Use one-one ness of f).
Page No 19:
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.
Page No 19:
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.
Page No 19:
Question 14:
Let
be a function defined as
. The inverse of f is map g: Range
(A) 
(B) 
(C) 
(D) 
Page No 25:
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)
* | 1 | 2 | 3 | 4 | 5 |
1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 1 | 2 | 1 |
3 | 1 | 1 | 3 | 1 | 1 |
4 | 1 | 2 | 1 | 4 | 1 |
5 | 1 | 1 | 1 | 1 | 5 |
Page No 25:
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 Exercise 4 above? Justify your answer.
Page No 25:
Question 6:
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find
(i) 5 * 7, 20 * 16 (ii) Is * commutative?
(iii) Is * associative? (iv) Find the identity of * in N
(v) Which elements of N are invertible for the operation *?
Page No 25:
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.
Page No 25:
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?
Page No 25:
Question 9:
Let * be a binary operation on the set Q of rational numbers as follows:
(i) a * b = a − b (ii) a * b = a2 + b2
(iii) a * b = a + ab (iv) a * b = (a − b)2
(v) 
(vi) a * b = ab2
Find which of the binary operations are commutative and which are associative.
a ∈ N.
But this relation is not true for any a ∈ N.
Thus, the operation * does not have any identity in N.
Page No 29:
Question 1:
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Page No 29:
Question 2:
Let f: W → W be defined as f(n) = n − 1, if 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.
Page No 29:
Question 4:
Show that function f: R → {x ∈ R: −1 < x < 1} defined by f(x) =
, x ∈R is one-one and onto function.
Page No 29:
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) =
)
Page No 29:
Question 7:
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and
Page No 29:
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 ⊂ B. Is R an equivalence relation on P(X)? Justify you answer:
Page No 30:
Question 9:
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
Page No 30:
Question 12:
Consider the binary operations*: R ×R → and o: R × R → R defined as 
and a o b = a, &mnForE;a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c ∈ R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Page No 30:
Question 15:
Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g: A → B be functions defined by f(x) = x2 − x, x ∈ A and
. Are f and g equal?
Justify your answer. (Hint: One may note that two function f: A → B and g: A → B such that f(a) = g(a) &mnForE;a ∈A, are called equal functions).
Page No 31:
Question 18:
Let f: R → R be the Signum Function defined as
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
, show that f o f(x) = x, for all
. What is the inverse of f?
, then fof(x) is
(B) x3 (C) x (D) (3 − x3)
