Animated Solution for Mathematics - Basic Mathematics: If $p\to (p\wedge \sim q)$ is false. Then the truth values of $p$ and $q$ are respectively

In the realm of mathematical logic, we are given the statement:

This is a structural failure in a logical argument. To solve this, we must work backward from the False result to determine the truth values of $p$ and $q$.

The heart of this problem lies in the implication operator $\to$. An implication $A\to B$ is only False if the antecedent $A$ is True and the consequent $B$ is False.

By mapping our statement to the form $A\to B$, we identify:

Since the entire statement is False, we are locked into the only possible scenario: $p$ must be True, and $(p\wedge \sim q)$ must be False. We have now determined with absolute certainty that $p=T$.

Next, we address the consequent:

We substitute our known value $p=T$ into this expression:

The conjunction (AND) operator is a strict gatekeeper; it only outputs True if both inputs are True. Since the output is False and the first input is True, the second input—the term $\sim q$—must be False.

We are left with the final piece of the puzzle:

The negation operator $\sim$ is an inverter. If the negation of $q$ is False, then $q$ itself must be the opposite. Therefore, $q=T$.

Through the rigorous application of logical rules, we have determined that both $p$ and $q$ are True. The final result is $(T,T)$.