Past paper Discrete maths

[fallen sable]

im trying to work this out but i got stuck idk what to do

[fallen sable]

im cooked

[hoary heart]

Given:
$$\exists x , (X \cdot P \lor \lnot Q)$$
Let $x_0$ be such that $X \cdot P \lor \lnot Q$.

\textbf{Case 1}: $X \cdot P$ is true for $x_0$.
Then $P$ is true for $x_0$.
Thus, $\lnot (X \cdot Q) \lor P$ holds for $x_0$ since $P$ is true.

\textbf{Case 2}: $\lnot Q$ is true for $x_0$.
Then $Q$ is false for $x_0$, so $X \cdot Q$ is false.
Thus, $\lnot (X \cdot Q)$ holds, and $\lnot (X \cdot Q) \lor P$ is true for $x_0$.

In either case, $\lnot (X \cdot Q) \lor P$ holds for $x_0$.

Hence:
$$\exists x , (\lnot (X \cdot Q) \lor P),$$
which is equivalent to:
$$\exists x , (X \cdot Q \implies P).$$

$\Box$