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Relations and Functions

Short Answer Type

231. Let R be a relation on the set of A of ordered pairs of positive integers defined by (x,y) R (u, v) if and only if x v = y u. Show that R is an equivalence relation.

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232.
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.

P(X) = { A : A is a subset of X }
(i) Since A C A ∀ ∈ P(X)
∴ ARA ∀ A ∈ P(X)
∴ R is reflexive relation (ii) Let A, B, C ∈ P(X) such that ARB and BRC A ⊂ B and B ⊂ C ⇒ A ⊂ C
⇒ A R C
∴ ARB and BRC ⇒ ARC
∴ R is transitive relation.
(iii) Now if A ⊂ B, then B may not be a subset of A i.e. ARB ⇏ BRA
∴ R is not a symmetric relation.
From (i), (ii), (iii), it follows that R is not an equivalence relation on P(X).

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233. 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 points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

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Long Answer Type

234.

Show that the relation R in the set A = { 1, 2, 3, 4, 5 } given by

R = { (a, b) : | a – b | is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

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Short Answer Type

235. Show that the number of equivalence relations in the set {1, 2, 3} containing (1, 2) and (2,1) is two.

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236. If R is the relation in N x N defined by (a, b) R (c, d) if and only if a + d = b + c, show that R is equivalence relation.

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237. In N x N, show that the relation defined by (a, b) R (c, d) if and only if a d = b c is an equivalence relation.

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238. Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d ) ⇔ a d (b + c) = b c (a + d). Check whether R is an equivalence relation on N x N.

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239.

For $straight a over straight b comma space c over d space element of space Q.$ the set of relational numbers, define $straight a over straight b space straight R space straight c over straight d$ if and only, if a d = b c. Show that R is an equivalence relation on Q.