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Question
Define the relation R in the set N × N as follows:
For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.
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Solution
Let (a, b), (c, d) ∈ N × N.
Then we have ab = ba .....(By the commutative property of multiplication of natural numbers)
⇒ (a, b) R (a, b)
Hence, R is reflexive.
Let (a, b), (c, d) ∈ N × N such that (a, b) R (c, d).
Then ad = bc
⇒ cb = da ......(By the commutative property of multiplication of natural numbers)
⇒ (c, d) R (a, b)
Hence, R is symmetric.
Let (a, b), (c, d), (e, f) ∈ N × N such that (a, b) R (c, d) and (c, d) R (e, f).
Then ad = bc, cf = de
⇒ adcf = bcde
⇒ af = be
⇒ (a, b) R (e, f)
Hence, R is transitive.
Since, R is reflexive, symmetric and transitive, R is an equivalence relation on N × N.
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