Question

Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive.

Answer 1

$\text{\bf{Symmetric but neither reflexive nor transitive}}$

Let A = {3,4,5}
Define a relation R on A as R = {(3,4), (4,3)}
Relation R is not reflexive as (3,3), (4,4) and (5,5) ∉ R.
Now, as (3,4) ϵ R and also (4,3) ϵ R,
R is symmetric.
⇒ (3,4), (4,3) ϵ R, but (3,3) ∉ R
⇒ R is not transitive.
Therefore, relation R is symmetric but not reflexive or transitive.

$\text{\bf{Transitive but neither reflexive nor symmetric.}}$
Let a relation R in R defined as:
R = {(a,b): a<b}
For any a ϵ R, we have (a,a)) ∉ R as a cannot be strictly less than a itself.
In fact, a = a,
Therefore, R is not reflexive.
Now, (1,2) ϵ R but 2 > 1
⇒ (2,1)) ∉ R.
⇒ R is not symmetric.
Now, let (a,b), (b,c) ϵ R
⇒ a < b and b < c
⇒ a < c
⇒ (a,c) ϵ R
⇒ R is transitive.
Therefore, relation R is transitive but not reflexive and a symmetric.

$\text{\bf{Reflexive and symmetric but not transitive.}}$
Let us take A = {2,4,6}
Define a relation R on A as:
A = {(2,2), (4,4), (6,6), (2,4), (4,2), (4,6), (6,4)}
Relation of R is reflexive as
for every a ϵ A,
(a,a) ϵ R
⇒ (2,2), (4,4), (6,6) ϵ R,
Relation R is symmetric as (a,b) ϵ R
⇒ (b,a) ϵ R for all a ,b ϵ R
And Relation R is not transitive as (2,4), (4,6) ϵR,
but (2,6) ∉ R

Therefore, relation R is reflexive and symmetric but not transitive.

$\text{\bf{Reflexive and transitive but not symmetric.}}$
Let us define a relation R in R as
R = {(a,b) : a3 ≥ b3}
It is clear that (a,a) ϵ R as a3 = a3
⇒ R is reflexive.
Now, (2,1) ϵ R
But (1,2) ∉ R
⇒ R is not symmetric.
Now, let (a,b) (b,c) ϵ R
⇒ a3≥ b3 and b3≥ c3
⇒ a3≥ c3
⇒ (a,c) ϵ R
⇒ R is transitive.

Therefore, relation R is reflexive and transitive but not symmetric.

$\text{\bf{Symmetric and transitive but not reflexive.}}$
Let A = {-7, -8}
Define a relation R on A as:
R = {(-7, -8), (-8, -7), (-7, -7)}
Relation R is not reflexive as (-8, -8) ∉ R
Relation R is symmetric as (-7, -8) ϵ R and (-8, -7) ϵ R
But it is seen that (-7, -8), (-8, -7) ϵ R.
Also, (-7, -7) ϵ R.
⇒ R is transitive.

Therefore, relation R is symmetric and transitive but not reflexive.