hello could you check an correct me please?
we'll say that $\Sigma$ follows property "a" if for every $a,b \in \Sigma$: $a \Rightarrow b$ or $b \Rightarrow a$.
1)I is a set of phrases that is defined: I consists of a single elemntary proposition p, and the only logical sign that is defined is $\Rightarrow$. show that set I is following "a" (i.e, for every $a,b \in \Sigma$: $a \Rightarrow b$ or $b \Rightarrow a$)
2)show that if $\Sigma$ follows "a", and each proposition in $\Sigma$ has a model in which it is true, then there's a model for $\Sigma$
my attempt:
1)building a building set: $a,b, b \Rightarrow a,a \Rightarrow b$we'll assume that it holds true for the base case and walk over $\lnot a$. if it follows property a, then if $a,b∈Σ: a⇒b$ or $b⇒a.$, but double contradiction we obtain it,for $\lor$ it is already defined, and in the case of $\land$, using the morgan rules and negation we can achieve the same.
regarding 2 i don't know how to prove it, but i believe it is derived using the structural induction. would appreciate your help with it.
thanank you very much for helping and correcting me.