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प्रश्न
The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .
विकल्प
reflexive but not symmetric
reflexive and transitive but not symmetric
an equivalence relation
none of the these
उत्तर
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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