The image you uploaded contains a set of exercises on propositional logic using algebraic laws. I will help you with the solutions step-by-step.

Here is the first problem:

1. $(p \rightarrow q) \land (p \rightarrow r) \equiv p \rightarrow (q \land r)$

Let's solve this using logical equivalences:

• $(p \rightarrow q)$ can be rewritten as $(\neg p \lor q)$.
• $(p \rightarrow r)$ can be rewritten as $(\neg p \lor r)$.

Thus, the left-hand side (LHS) of the expression becomes: $(\neg p \lor q) \land (\neg p \lor r)$

Now, using the distributive property of logical conjunction: $(\neg p \lor q) \land (\neg p \lor r) \equiv \neg p \lor (q \land r)$

This expression is logically equivalent to $p \rightarrow (q \land r)$, since $\neg p \lor (q \land r)$ is the definition of the implication $p \rightarrow (q \land r)$.

Therefore, the expression $(p \rightarrow q) \land (p \rightarrow r) \equiv p \rightarrow (q \land r)$ is true.

Would you like me to continue with the other problems?