Let R be a relation on N x N defined by (a, b) R (c, d)a + d = b + c f

1.	Let R be a relation on N x N defined by (a, b) R (c, d)a + d = b + c for all (a, b), (c, d)N x N. Show that :i. (a, b) R (a, b) for all (a, b) N x N ii. (a, b) R (c, d) (c, d) R (a, b) for all (a, b), (c, d) N x N iii. (a, b) R (c, d) and (c, d) R (e, f)(a, b) R (e, f) for all (a, b), (c, d), (e, f) N × N
Answer» Given, (a, b) R (c, d) a + d = b + c for all (a, b), (c, d)N x N i. (a, b) R (a, b) ⇒ a + b = b + a for all (a, b) N x N ∴ (a, b) R (a, b) for all (a, b) N x N ii. (a, b) R (c, d) ⇒ a + d = b + c ⇒ c + b = d + a ⇒ (c, d) R (a, b) for all (c, d), (a, b) N x N iii. (a, b) R (c, d) and (c, d) R (e, f) a + d = b + c and c + f = d + e ⇒ a + d + c + f = b + c + d + e ⇒ a + f = b + c + d + e – c – d ⇒ a + f = b + e ⇒ (a, b) R (e, f) for all (a, b), (c, d), (e, f) N × N

Let R be a relation on N x N defined by (a, b) R (c, d)a + d = b + c for all (a, b), (c, d)N x N. Show that :i. (a, b) R (a, b) for all (a, b) N x N ii. (a, b) R (c, d) (c, d) R (a, b) for all (a, b), (c, d) N x N iii. (a, b) R (c, d) and (c, d) R (e, f)(a, b) R (e, f) for all (a, b), (c, d), (e, f) N × N

Answer»

Given,

(a, b) R (c, d) a + d = b + c for all (a, b), (c, d)N x N

i. (a, b) R (a, b)

⇒ a + b = b + a for all (a, b) N x N

∴ (a, b) R (a, b) for all (a, b) N x N

ii. (a, b) R (c, d)

⇒ a + d = b + c

⇒ c + b = d + a

⇒ (c, d) R (a, b) for all (c, d), (a, b) N x N

iii. (a, b) R (c, d) and (c, d) R (e, f)

a + d = b + c and c + f = d + e

⇒ a + d + c + f = b + c + d + e

⇒ a + f = b + c + d + e – c – d

⇒ a + f = b + e

⇒ (a, b) R (e, f) for all (a, b), (c, d), (e, f) N × N

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