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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.

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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 ?

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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.

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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 ∧.

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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.

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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 ?

A = N x N and (a, b) * (c, d) = (ad + bc, bd)
(i) Let (a, b), (c, d), (e,f) be any three elements of A.
∴ {(a, b) * (c, d)} * (e,f) = (ad + bc, bd) * (e, f)
= ((ad + bc) f + (bd) e, (bd) f)
∴ {(a, b) * (c, d)} * (e,f) = (adf + btf + bde, bdf)    ...(1)
Again (a, b) * {(c, d) * (e,f)} = (a, b) * (c f + de, d f)
= (a(d f) + b (c f + de), b (d f))
∴ (a, b) * {(c, d) * (e,f)} = (adf + bef + bde, bdf)    ...(2)
From (1) and (2), we get,
(a, b) * {(c, d) * (e,f)} = {(a, b) * (c, d) * (e,f)} ∴ (A. *) is associative.
(ii)    If possible, suppose that (x, y) is identity element in A.
∴ (a, b) * (x, y) = (a, b) ∀ (a, b) ∈ A
⇒ (ay + bx by) = (a, b) ∀ (a, b) ∈ A ⇒ ay + bx = a and by = b ∀ a, b ∈ N ⇒ x = 0, y = 1 ∀ a, b ∈ N This is not possible as 0 ∉ N ∴ our supposition is wrong ∴ (A, *) has no identity element.
(iii)    Let (a, b), (c, d) be any two elements of A.
Now (a, b) *(c, d) = (ad + be , bd)
= (bc + ad,db)
= (cb + da , db)
∴ (a, b) * (c, d)) = (c, d) * (a, b) ∀ (a, b), (c, d) ∈A ∴ (A, *) is commutative .

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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).

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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.

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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 *.

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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.

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