i am not sure regarding those two claims i've solved. would appreciate your comments and corrections to learn better and improve. it's basically a one questions devided into prove/disprove with two claims each.
the questions:
1)let $\Sigma_1$ $\Sigma_2$ be sets of propositions.
a)$\Sigma_1\cup\Sigma_2$ are true in some model
b)for every proposition $a$ so that $\Sigma_1 \Rightarrow a$ $\Sigma_2 \Rightarrow b$ , $a\land b$ has a model
2)the following two claims are equivalent:
a)a is a tautology
b)for every proposition b and y, ${y} \Rightarrow b \iff {a,y} \Rightarrow b$ (for some reason it eliminates the {}, so it should be {y} -> b <-> {a,y} -> b
what i tried to do:
1)not true. if $\Sigma_1 \cup \Sigma_2$ is real in some model,then it doesn't neccesarily mean that for every proposition $a$ so that $\Sigma_1 \Rightarrow a$ $\Sigma_2 \Rightarrow b$ - $a\land b$ has a model. i even tried to play a lot with de morgans laws($¬(∧)≡(¬)∨(¬)$, $¬(∨)≡(¬)∧(¬)$ and couldn't find a relation that will hold true).
2)true. if y yields b and a is a tautology, then {a,y} yields b, whih makes both claims hold true.
please correct me if i've done mistakes so i can learn and improve, and show me the right way if i'e done it inappropriately
Regarding your first question, you can easily find a model of $a\land b$: in fact, a model of $\Sigma_1\cup \Sigma_2$ is a model of $a\land b$. You may need a precise definition of a model and satisfaction relation to give a formal, precise proof.
Here is an example of a formal proof: I will assume you work over the propositional logic with truth values. Assume that $v$ is a truth assignment (that is, a function which sends propositions to its truth value) which validates $\Sigma_1\cup\Sigma_2$. Then we have $v(a)=1$ and $v(b)=1$ since $\Sigma_1\models a$ and $\Sigma_2\models b$. Hence $v(a\land b) = 1$.
Your guess at the second question is right. To prove the other direction of the second formally, take $y$ as a tautology and $b$ as an absurdity.