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Question
The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .
Options
reflexive but not symmetric
reflexive and transitive but not symmetric
an equivalence relation
none of the these
Solution
an equivalence relation
We observe the following properties of relation R.
Reflexivity: Let (a, b) ∈ N × N
⇒ a, b ∈ N
⇒ a+b = b+a
⇒ (a, b) ∈ R
So, R is reflexive on N×N.
Symmetry: Let (a, b), (c, d) ∈ N × N such that (a, b) R (c, d)
⇒ a+d = b+c
⇒ d+a = c +b
⇒ (d, c), (b, a) ∈ R
So, R is symmetric on N×N.
Transitivity : Let (a, b), (c, d), (e, f) ∈ N×N such that (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 + e
⇒(a, b) R (e, f)
So, R is transitive on N×N.
Hence, R is an equivalence relation on N.
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