Mark the tick against the correct answer in the following:
Let R be a relation on N × N, defined by
(a, b) R (c, d) ⇔ a + d = b + c. Then, R is
According to the question ,
R = {(a, b) , (c, d) : a + d = b + c }
Formula
For a relation R in set A
Reflexive
The relation is reflexive if (a , a) ∈ R for every a ∈ A
Symmetric
The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R
Transitive
Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R
Equivalence
If the relation is reflexive , symmetric and transitive , it is an equivalence relation.
Check for reflexive
Consider , (a, b) R (a, b)
(a, b) R (a, b) ⇔ a + b = a + b
which is always true .
Therefore , R is reflexive ……. (1)
Check for symmetric
(a, b) R (c, d) ⇔ a + d = b + c
(c, d) R (a, b) ⇔ c + b = d + a
Both the equation are the same and therefore will always be true.
Therefore , R is symmetric ……. (2)
Check for transitive
(a, b) R (c, d) ⇔ a + d = b + c
(c, d) R (e, f) ⇔ c + f = d + e
On adding these both equations we get , a + f = b + e
Also,
(a, b) R (e, f) ⇔ a + f = b + e
∴ It will always be true
Therefore , R is transitive ……. (3)
Now , according to the equations (1) , (2) , (3)
Correct option will be (D)