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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
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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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