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The characteristic of sets {1, 4, 7 }, {2, 5, 8} and {3, 6, 9} is that difference between any two elements of these sets is a multiple of 3. ∴ (x,y) ∈ R 1 ⇒ x – y is a multiple of 3 ⇒ {x,y} ⊂ {1,4,7} or {x, y} ⊂ {2, 5, 8} or {x,y} ⊂ {3, 6, 9} ⇒ (x,y) ∈ R 2 . Hence R 1 ⊂ R 2 . Similarly, { x, y} ∈ R 2 ⇒ {x, y} ⊂ {1, 4, 7} or {x,y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ x – y is divisible by 3 ⇒ {x ,y} ∈ R 1 . ∴ R 2 ⊂ R 1 . Hence, R 1 = R 2 .

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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.

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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.

The characteristic of sets {1, 4, 7 }, {2, 5, 8} and {3, 6, 9} is that difference between any two elements of these sets is a multiple of 3.
∴ (x,y) ∈ R₁ ⇒ x – y is a multiple of 3
⇒ {x,y} ⊂ {1,4,7} or {x, y} ⊂ {2, 5, 8} or {x,y} ⊂ {3, 6, 9}
⇒ (x,y) ∈ R₂.
Hence R₁ ⊂ R₂.
Similarly, { x, y} ∈ R₂⇒ {x, y} ⊂ {1, 4, 7} or {x,y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ x – y is divisible by 3 ⇒ {x ,y} ∈ R₁.
∴ R₂ ⊂ R₁.
Hence, 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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