# logical implication

• Jun 13th 2011, 02:22 PM
alexandros
logical implication
Does $X\subseteq Y$ logically imply xεX and xεY ??
• Jun 13th 2011, 02:29 PM
Plato
Re: logical implication
Quote:

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 $x\notin X\text{ or }x\in Y$.
• Jun 13th 2011, 02:45 PM
emakarov
Re: logical implication
I would even start by saying that x has not been defined...
• Jun 13th 2011, 03:42 PM
alexandros
Re: logical implication
Yes i agree with both of you,but how do you show that to somebody who insists otherwise??
• Jun 13th 2011, 04:01 PM
Plato
Re: logical implication
Quote:

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 $\left( {\forall x} \right)\left[ {x \in X \wedge x \in Y} \right]$
• Jun 13th 2011, 04:09 PM
alexandros
Re: logical implication
O.k Thanks we carry on tomorrow