Animated Solution for Mathematics - Sets and Relations: If $(P\wedge \sim q)\wedge (p\wedge r)\to \sim p\vee q$ is false, then the truth values of $p,q$ and $r$ are respectively

The expression provided is $(p\wedge \sim q)\wedge (p\wedge r)\to \sim p\vee q$. We are given that this entire logical statement evaluates to False.

In formal logic, an implication $A\to B$ is False if and only if the antecedent $A$ is True and the consequent $B$ is False.

This gives us two critical conditions: 1. The antecedent $(p\wedge \sim q)\wedge (p\wedge r)$ must be True. 2. The consequent $\sim p\vee q$ must be False.

We begin with the consequent, $\sim p\vee q$, because it provides the most restrictive constraints. An OR ($\vee$) operation is False only when both components are False.

Therefore, we must have: 1. $\sim p=F\Rightarrow p=T$ 2. $q=F$

By analyzing the consequent, we have already determined that $p$ is True and $q$ is False.

Now we examine the antecedent: $(p\wedge \sim q)\wedge (p\wedge r)$. For this conjunction to be True, both $(p\wedge \sim q)$ and $(p\wedge r)$ must be True.

Let us verify the first part with our known values: This confirms our values for $p$ and $q$ are consistent with the antecedent.

Now, we evaluate the second part: $(p\wedge r)$. Since we know $p=T$, the expression becomes: For this to hold true, $r$ must be True.

By methodically applying the rules of logical operators, we have isolated the truth values for each variable.

The values that satisfy the condition are: $p=T$ $q=F$ $r=T$

The final sequence of truth values is $(T,F,T)$.