What English sentences are represented by following expressions? Let S = Steve is happy, G = George is happy
$1. (S \lor G) \wedge (\lnot S \lor (\lnot G)) $
One of Steve or George is happy and other is sad
$2. [S \lor (G \wedge (\lnot S)] \lor (\lnot G)$
George is happy when Steve is sad or George is sad
$3. S \lor [G \wedge (\lnot S \lor \lnot G) ]$
Steve is sad when George is happy or Steve is happy
Can anyone cross verify my solutions.Thank you
 A: As we define that not happy if and only if sad, we have $1.$ traslated correctly. But there are some ambiguity for the second one, for example

(George is happy when Steve is sad) or George is sad $($Correct translation for $2.)$

$$(\neg S\to G) \lor \neg G\tag{1}$$

George is happy when (Steve is sad or George is sad)

$$(\neg S\lor\neg G)\to G\tag{2}$$
The second expression $[S∨(G∧(¬S)]∨(¬G)$ is actually equivalent to $(1)$ which says "(George is happy when Steve is sad) or George is sad", but not $(2)$. Consider a counter example $G=S=0$, that $$(¬S→G)∨¬G=(¬S∨¬G)→G$$
$$(¬0→0)∨¬0=(¬0∨¬0)→0$$
$$0∨1=1→0$$
$$1=0$$
This proves they are not the same when George and Steve are both sad.
Similarly for the third one

(Steve is sad when George is happy) or Steve is happy

$$(G\to S)\lor S\tag{3}$$

Steve is sad when (George is happy or Steve is happy)

$$(G\lor S)\to S\tag{4}$$
However, interestingly this time $(3)$ and $(4)$ are equivalent since
\begin{align}
(G\to S)\lor S\equiv&\neg G\lor S\lor S\\
\equiv&\neg G\lor S\\
\equiv&\neg G\lor S\land \top\\
\equiv&(\neg G\lor S)\land(\neg S\lor S)\\
\equiv&(\neg G\land\neg S)\lor S\\
\equiv&(G\lor S)\to S
\end{align}
And the expression $(S∨(G∧(¬S∨¬G)))$ is equivalent to $\neg S\to G$ which says that

George is happy when Steve is sad. $($Correct translation for $3.)$

A: 1
Your answer is correct.
2
\begin{align} [∨(∧¬)]∨¬       & \equiv  [(∨)∧(∨¬)]∨¬ \\       & \equiv [(∨)∧\top]∨¬ \\        & \equiv (∨)∨¬ \\            & \equiv ∨(∨¬) \\          & \equiv ∨\top \\            & \equiv \top       \end{align}
Therefore, the proposition is a tautology, and

Steve and George can have any mood.

3
\begin{align}
S∨[G∧(¬S∨¬G)] & \equiv S∨[(G∧¬S)∨(G∧¬G))] \\
&\equiv S∨[(G∧¬S)∨\bot)] \\
&\equiv S∨(G∧¬S) \\
&\equiv (S∨G)∧(S∨¬S) \\
&\equiv (S∨G)∧(S∨¬S) \\
&\equiv (S∨G)∧\top \\
&\equiv S∨G \\
&\end{align}

Either Steve or George is happy, could be both.

