Help me with appropriate answer..

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Relations and Functions

Short Answer Type

301. 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.

124 Views

302. 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 ?

271 Views

Long Answer Type

303.

Let A = N x 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.

109 Views

Short Answer Type

304.

Consider the binary operation A on the set {1, 2, 3, 4, 5} defined by a ∧ b = min. {a, b}. Write the operation table of the operation ∧.

114 Views

305.

Let A = N x N and let ‘*’ be a binary operation on A defined by (a, b) * (c, d) = (a c, b d). Show that
(i) (A, *) is associative (ii) (A, *) is commutative.

194 Views

Long Answer Type

306.

Let A = N x N and let ‘*’ be a binary operation on A defined by
(a, b) * (c, d) = (ad + bc, bd). Show that
(i) (A, *) is associative, (ii) (A, *) has no identity element,
(iii) Is (A, *) commutative ?

162 Views

307.

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication 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

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative ?
(iii) Compute (2 * 3) * (4 * 5).

123 Views

308. Consider the binary operations * : R x R → R and a : R x R → R defined as a * b = | a – b | and a o b = a ∀ a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ 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.

a * b = | a – b |
= | – (b – a) = | b – a | = b * a
⇒ a * b = b * a ∀ a, b ∈ R
⇒ ‘*’ is commutative.
Let a, b, c ∈ R,
∴ (a * b) * c = | a – b | * c = || a – b | – c |
and a * (b * c) = a * | b – c | = | a – | b – c||
⇒    (a * b) * c ≠ a * (b * c)
∴ ‘*’ is not associative.
Again    a o b = a and b o a = b
⇒    a o b ≠ b o a
⇒ o is not commutative.
Now, let a, b, c ∈R
∴ (a o b) o c = a o c = a and a o(b o c) = a o b = a ⇒ ‘o’ is associative.
Again a * (b o c) = a * b = | a – b | and (a * b) o (a * c) = | a – b | o | a – c |
= | a – b | ∴ a * (b o c) = (a * b) o (a * c)
Also a o(b * c) = a o | b – c | = a and (a o b) * (a o c) = a * a = | a – a | = 0 ⇒    a o (b * c) ≠ (a o b) *(a o c)
∴ ‘o’ is not distributive over ‘*’.

158 Views

Short Answer Type

309. Given a non-empty set X, consider the binary operation * : P(X) x P(X) → P(X) given by A * B = A ∩ 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 only invertible element in P(X) with respect to the operation *.

122 Views

310. Given a non-empty set X, let * : P(X) x P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set ϕ is the identity for the operation * and all the elements A of P(X) are invertible with A^–1 = A.

134 Views