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The smallest relation R 1 containing (1, 2) and (2, 3) which is reflexive and transitive but not symmetric is {(1, 1), (2. 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Now, if we add the pair (2. 1) to R 1 to get R 2 , then the relation R 2 will be reflexive, transitive but not symmetric. Similarly, we can obtain R 3 and R 4 by adding (3, 2) and (3, 1) respectively, to R 1 to get the desired relations. However, we can not add any two pairs out of (2, 1), (3, 2) and (3, 1) to R 1 at a time, as by doing so, we will be forced to add the remaining third pair in order to maintain transitivity and in the process, the relation will become symmetric also which is not required. the total number of desired relations is four.

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Relations and Functions

Short Answer Type

211. Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

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212. 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.

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213. Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.

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214. If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’.

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215. If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

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216. Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

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217. Let A = {1. 2. 3}. Then show that the number of relations containing (1,2) and (2. 3) which are reflexive and transitive but not symmetric is four.

The smallest relation R₁ containing (1, 2) and (2, 3) which is reflexive and transitive but not symmetric is {(1, 1), (2. 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Now, if we add the pair (2. 1) to R₁to get R₂, then the relation R₂ will be reflexive, transitive but not symmetric. Similarly, we can obtain R₃ and R₄ by adding (3, 2) and (3, 1) respectively, to R₁ to get the desired relations. However, we can not add any two pairs out of (2, 1), (3, 2) and (3, 1) to R₁ at a time, as by doing so, we will be forced to add the remaining third pair in order to maintain transitivity and in the process, the relation will become symmetric also which is not required. the total number of desired relations is four.

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218. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R₁ be a relation in X given by R₁ = {(x, y) : x – y is divisible by 3} and R₂ be another relation on X given by R₂ = {(x, y) : {x, y} ⊂ {1,4, 7 }} or ⊂ {2, 5, 8} or {x, y,}⊂ {3, 6, 9}}. Show that R₁ = R₂.

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219.

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

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220.

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

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