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

$\text{\bf{reflexive relation}}$ A relation is said to be reflexive relation, if every element of A is related to itself.
thus,(a,a) ∈ R , for all a∈ A => R is reflexive

$\text{\bf{symmetric relation}}$ A relation is said to be symmetric relation, if
(a, b) ∈ R => (b, a) ∈ R , for all a, b ∈ A
e.g., aRb => bRa, for all a, b ∈ A => R is symmetric.

$\text{\bf{transitive relation}}$ A relation is said to be transitive relation if
(a, b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a,b,c ∈ A

now let's check out all relations :
Let us take A = {1, 2, 3, 4, 5, 6}
A relation R is defined on set A as:
R = {(a, b): b = a + 1}
Then, R = {(1,2), (2,3), (3,4), (4,5), (5,6)}
Now, we will find (a, a) ∉ R, where a ϵ A
For instance,
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) ∉ R
Therefore, R is not reflexive.
We can see that (1,2) ϵ R, but (2,1) ∉ R.
Therefore, R is not symmetric.
And now, (1,2), (2,3) ϵ R
But, (1,3) ∉ R
Therefore, R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.