# Thread: R U S, reflexive (Proof Method)

1. ## R U S, reflexive (Proof Method)

Hello,

Suppose that r and s are reflexive relations on a set A. Prove or Disprove this statement.

r U s is reflexive

I'm stuck with proof by contradiction

2. Originally Posted by l flipboi l
Suppose that r and s are reflexive relations on a set A. Prove or Disprove this statement.
If $\displaystyle \mathcal{R}$ is a reflexive relation on set $\displaystyle A$ then $\displaystyle \Delta _A \subseteq \mathcal{R}$.
If $\displaystyle \mathcal{S}$ is any other relation on $\displaystyle A$ then
$\displaystyle \Delta _A \subseteq \mathcal{R}\cup \mathcal{S}$ which means the union is reflexive.

3. Originally Posted by Plato
If $\displaystyle \mathcal{R}$ is a reflexive relation on set $\displaystyle A$ then $\displaystyle \Delta _A \subseteq \mathcal{R}$.
If $\displaystyle \mathcal{S}$ is any other relation on $\displaystyle A$ then
$\displaystyle \Delta _A \subseteq \mathcal{R}\cup \mathcal{S}$ which means the union is reflexive.
Thanks! is this direct proof?

4. Originally Posted by l flipboi l
is this direct proof?
That proof is about as direct as it ever gets.

5. Originally Posted by Plato
That proof is about as direct as it ever gets.
Thanks! is there a way to show using proof by contradiction?

6. Originally Posted by l flipboi l
Thanks! is there a way to show using proof by contradiction?
Yes, but then we end up using the very idea I gave you in the so called direct proof.