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

R = {(a, b) : a ≤ b}
(i) Since (a, a) ∈ R ∀ a ∈ R [∵ a ≤ a ∀ a ∈ R]
∴ R is reflexive.
(ii) (a, b) ∈ R ⇏ (b, a) ∈ R [∵ if a ≤ b. then b ≤ a is not true]
∴ R is not symmetric.
(iii) Let (a, b), (b, c) ∈ R ∴ a ≤ b, b ≤ c ∴ a ≤ c ⇒ (a, c) ∈ R ∴ (a, b), (b. c) ∈ R ⇒ (a, c) ∈ R ∴ R is transitive

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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.

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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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