Find a Logical Formula using five atomic sentences that is true Exactly when Three out of Five sentences are true.

Introduction:

Consider a scenario where we have five atomic sentences, each of which can be either true or false. Our goal is to find a logical formula that evaluates to true only when exactly three out of

these five sentences are true. To approach this problem systematically, we will first construct a truth table with $2^5$ rows representing all possible combinations of truth values for the five input

sentences. Then, we'll analyze this truth table to derive the formula in normal conjunctive form.

Hint:

To find a formula that evaluates to true only when exactly three out of the five atomic sentences are true, focus on the rows in the truth table where exactly three inputs are true and the rest are

false. Observe the patterns in these rows to derive the logical formula in normal conjunctive form.