R U S, reflexive (Proof Method)

l flipboi l

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 (Thinking)

Plato

MHF Helper
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.

l flipboi l

l flipboi l

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?

Plato

MHF Helper
is this direct proof?
That proof is about as direct as it ever gets.

l flipboi l

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

Plato

MHF Helper
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.