Question

Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

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 all relations :
It is given that R = {(a, b): a ≤ b},
It is clear that (a, a) ϵ R as a = a.
Therefore, R is reflexive.
Now let us take (2,4) ϵ R (2 < 4)
But, (4,2) ∉ R as 4 is greater than 2.
Therefore, R is not symmetric.
Now, let (a, b), (b, c) ϵ R
Then, a ≤ b and b ≤ c
⇒ a ≤ c
⇒ (a, c) ϵ R
Therefore, R is a transitive.
Therefore, R is reflexive and transitive but not symmetric.