Equivalent sets of wffs

Question

Propositional logic. Given two sets of wffs, $\Sigma$ and $\Gamma$, are the following definitions of equivalence between $\Sigma$ and $\Gamma$ .....equivalent?

1. $(\Sigma\vDash\Gamma)\land(\Gamma\vDash\Sigma)$

2. For all $\alpha$, $(\Sigma\vDash\alpha)\leftrightarrow(\Gamma\vDash\alpha)$

I'm not looking for a proof, a just want to avoid engaging myself in a lost battle.

Answer 1

Two sets of formulas are equivalent, if any formula of the one set is a consequence of the other and conversely.

Equivalently, they have the same models.

This means :

$\text {for every } \alpha \in \Gamma : \Sigma \vDash \alpha$ and $\text { for every } \beta \in \Sigma : \Gamma \vDash \beta$.