# Logical equivalence and simplifications for $s\land[\neg(\neg s \lor t)]\lor(s \land t)$ and $\neg(s \lor \neg t)\lor (\neg s \land \neg t)$

Are the steps I applied correct in simplifying logical statements and producing logical equivalencies?

(1) Simplify $s\land(\lnot t\lor s):$ \begin{align} s\land(\lnot t\lor s) &\Leftrightarrow s & \text{by absorption} \\ \end{align}

(2) Simplify $s\land[\neg(\neg s \lor t)]\lor(s \land t):$ \begin{align} s\land[\neg(\neg s \lor t)]\lor(s \land t) &\Leftrightarrow s\land[\neg(\neg s)\land \neg t]\lor (s\land t) & \text{by DeMorgan's}\\ &\Leftrightarrow s\land(s \land \neg t)\lor (s\land t) & \text{by double negation}\\ &\Leftrightarrow [(s\land s) \land \neg t]\lor (s\land t) &\text{by associativity of $\land$} \\ &\Leftrightarrow (s\land\neg t)\lor(s\land t) &\text{by idempotence}\\ &\Leftrightarrow s \land(\neg t\lor t) & \text{by distributivity of $\land$ over $\lor$}\\ &\Leftrightarrow s\land T &\text{by inverse}\\ &\Leftrightarrow s & \text{by identity} \end{align}

(3) Simplify $\neg(s \lor \neg t)\lor (\neg s \land \neg t):$ \begin{align} \neg(s \lor \neg t)\lor (\neg s \land \neg t) &\Leftrightarrow \neg s\lor\neg(\neg t)\lor(\neg s\land\neg t) & \text{by DeMorgan's}\\ &\Leftrightarrow \neg s\land t\lor(\neg s\land\neg t) & \text{by double negation}\\ &\Leftrightarrow (\neg s\land t) \lor(\neg s\land\neg t) &\text{} \\ &\Leftrightarrow \neg s\land(t\lor\neg t) &\text{by distributivity of $\land$ over $\lor$}\\ &\Leftrightarrow \neg s\land T& \text{by inverse}\\ &\Leftrightarrow \neg s &\text{by identity}\\ \end{align}