For any propostional sentences $a,b,c$, if $a\models (b\wedge c)$, then $a\models b$ and $a\models c$

Question

I'm having a hard time dealing with the $\models$ symbol. I don't really know how to reason through or manipulate these equations to prove why or why not the result holds. Another similar question is:

For any propostional sentences $a,b,c$, if $a\models (b\vee c)$, then $a\models b$ and $a\models c$.

Help on either of these would be highly appreciated.

Answer 1

Regarding :

if $a \vDash (b∨c)$, then $a \vDash b$ and $a \vDash c$.

We have that : $a \vDash (b∨c)$ means that for every valuation $v$, if $v(a)=T$, then $v(b∨c)=T$.

But $v(b∨c)=T$ means that $v(b)=T$ or $v(c)=T$, but not necessarily both.

We can have $v(b)=T$ and $v(c)=F$ and this is enough for $v(b∨c)=T$.

So, it is not true that : if $a \vDash (b∨c)$, then $a \vDash b$ and $a \vDash c$.

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For :

if $a \vDash (b \land c)$, then $a \vDash b$ and $a \vDash c$,

we have that for every valuation $v$, if $v(a)=T$, then $v(b \land c)=T$, and this one means that $v(b)=T$ and $v(c)=T$.

Thus, in this case it is true that : if $a \vDash (b \land c)$, then $a \vDash b$ and $a \vDash c$.