Question

Use the Laws of Logic to prove the following equivalence and state the laws used for each step.

p → (p ⋀ q), p → q

Answer 1

We remind three Laws of Logic:

1. The law of identity: $P$ is $P$.
2. The law of noncontradiction: $P$ is not non-$P$
3. The law of the excluded middle: either $P$ or non-$P$

Using the third law we get that the statement $p$ is true or false. Suppose that $p$ is true. Then, $p\land q$ is true or false. Namely, if $q$ is false, then $p\land q$ is false. Otherwise, if $q$ is true, then $p\land q$ is true. If $p$ is false, then $p\land q$ is false. Implication sign $\rightarrow$ by definition means that if there is true on the left side, then it must be true on the right side and if it is false on the left side of $\rightarrow$, then it can be true or false on the right side. Thus, we received that $p\rightarrow p\land q$.