Final answer:
The formula p∧(p∧q) simplifies to p∧q by applying the associative property of conjunction and the identity property where a variable AND-ed with itself remains unchanged.
Step-by-step explanation:
To simplify the formula p∧(p∧q), we apply the logical conjunction rules. The conjunction (AND) operation ∧ is associative, which means that the grouping of the operands does not affect the result. In simpler terms, the expression p∧(p∧q) is equivalent to (p∧p)∧q, and since p∧p is always equal to p (because the truth of p will not change when AND-ed with itself), the entire expression simplifies to just p∧q.
Additionally, we can apply the identity property of the logical AND operation, where any variable AND-ed with itself remains unchanged. So there is no need to perform the operation twice as it would not change the outcome. Therefore, p∧(p∧q) simplifies directly to p∧q.