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

Solution:

a ≤ a: which is valid, (a, a) ∈ R, So R is reflexive.

a ≤ b yet b ≤ a (bogus): (a, b) ∈ R however (b, a) ∉ R, So R isn’t symmetric.

Once more, a ≤ b and b ≤ c then a ≤ c : (a, b) ∈ R and (b, c) and (a, c) ∈ R, So R is transitive. Thusly, R is reflexive and transitive however not symmetric.