Prove that Statements forms are tautologies Given variable statement forms $A$ and $B$. How to prove that if $(A\land B)$ is a tautology then $A$ and $B$ are tautologies too?.
Mi approach would be a proof by contradiction, something like: If we suppose $(A \land B)$ is a tautology but $A$ and $B$ are not tautologies, then $(A \land B)$ can't be a tautology and this will contradict the hypothesis, so $A$ and $B$ are tautologies. Is this correct or near correct?
 A: Your reasoning works, but it would be best to do this with more symbolic justification as to why "then $(A \land B)$ can't be a tautology.

...suppose [for the sake of contradiction) that $(A \land B)$ is a tautology but $A$ and $B$ are not [both] tautologies,

So $A \not\equiv T$ or $B\not\equiv T$ (or both).

then $(A \land B)$ can't be a tautology 

because...there would then be at least one truth value assignments such that $A$ is false, or $B$ is false, and hence there would be truth value assignments such that $A\land B$ is false. This means that $A \land B \not\equiv T$. That is, it is not true that  $A \land B$ is a tautology.

and this will contradict the hypothesis, so $A$ and $B$ [must both be] tautologies. 

A: Your approach seems correct but it uses we know that $A∧B=T$ if $A,B $ are tautologies.

I always like to see these using Venn-Diagrams. $A,B$ be two sets and $A∧B$ represent the intersection 


*

*Universal set=Tautology.

*empty ($\phi$ ) set represent contradiction.


. If $A∧B=\text{universal set}$ . Then both $A,B $ must be universal set, or tautologies.
A: The semantics of $\land$ is that
$$ \Phi \models \phi \land \psi \mbox{ if and only if } \Phi \models \phi \mbox{ and } \Phi\models\psi \mbox{.} \tag 1$$
A sentence $\phi$ is a tautology if and only if
$$\models\phi \mbox{.} \tag 2$$
Suppose that $A \land B$ is a tautology. Then, by $(2)$, $\models A \land B$. Then, by $(1)$, $\models A$ and $\models B$. Then, by $(2)$, $A$ and $B$ are tautologies.
