1. ## logical implication

Does $\displaystyle X\subseteq Y$ logically imply xεX and xεY ??

2. ## Re: logical implication

Originally Posted by alexandros
Does $\displaystyle X\subseteq Y$ logically imply xεX and xεY ??
Absolutely not!
It means $\displaystyle \text{If }x\in X\text{ then }x\in Y$.
Or equivalently $\displaystyle x\notin X\text{ or }x\in Y$.

3. ## Re: logical implication

I would even start by saying that x has not been defined...

4. ## Re: logical implication

Yes i agree with both of you,but how do you show that to somebody who insists otherwise??

5. ## Re: logical implication

Originally Posted by alexandros
Yes i agree with both of you,but how do you show that to somebody who insists otherwise??
In this matter I absolutely agree with emakarov's reply.
$\displaystyle \left( {X \subseteq Y} \right) \Leftrightarrow \left( {\forall x} \right)\left[ {x \in X \to x \in Y} \right]$.

Ask your protagonists does imply that $\displaystyle \left( {\forall x} \right)\left[ {x \in X \wedge x \in Y} \right]$

6. ## Re: logical implication

O.k Thanks we carry on tomorrow