Question

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. 

Answer 1

Let A = {1, 2, 3, 4, 5, 6}.
A relation R is defined on set A as:
R = {(a, b): b = a + 1}
∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
We can find (a, a) ∉ R, where a ∈ A.
For instance,
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R
∴R is not reflexive.

It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.
∴R is not symmetric.

Now, (1, 2), (2, 3) ∈ R
But, (1, 3) ∉ R
∴R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.