logical implication

**alexandros** · June 13th 2011, 02:22 PM

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

**Plato** · June 13th 2011, 02:29 PM

Originally Posted by alexandros

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

Absolutely not!
It means $\text{If }x\in X\text{ then }x\in Y$ .
Or equivalently $x\notin X\text{ or }x\in Y$ .

**emakarov** · June 13th 2011, 02:45 PM

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

**alexandros** · June 13th 2011, 03:42 PM

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

**Plato** · June 13th 2011, 04:01 PM

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.
$\left( {X \subseteq Y} \right) \Leftrightarrow \left( {\forall x} \right)\left[ {x \in X \to x \in Y} \right]$ .

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

**alexandros** · June 13th 2011, 04:09 PM

O.k Thanks we carry on tomorrow

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