Animated Solution for Mathematics - Sets and Relations: Among the statements $(S1):(p\Rightarrow q)\vee ((\sim p)\wedge q)$ is a tautology, $(S2):(q\Rightarrow p)\Rightarrow ((\sim p)\wedge q)$ is a contradiction

A Journey Through Truth Tables

Welcome, fellow explorers of the mathematical universe! Today, we are not just solving a problem; we are building a map.

In the realm of mathematical reasoning, we often encounter statements that seem complex, but beneath the surface, they are governed by the elegant, binary rhythm of True and False. We are tasked with analyzing two logical statements, $S1$ and $S2$, to determine if they are a tautology or a contradiction.

The Foundation of Truth

Before we dive into the complexity, let us define our tools. A tautology is a statement that is always True, no matter what the truth values of its components are. A contradiction is the exact opposite—a statement that is always False.

To test this, we use the truth table, a systematic grid that explores every possible reality. Since we have two basic propositions, $p$ and $q$, we have $2^2=4$ possible combinations.

We lay them out: (True, True), (True, False), (False, True), and (False, False). This is our canvas.

Deconstructing $S1$ (The Tautology Hunt)

Our first statement is:

To evaluate this, we need to break it down. First, we find $\sim p$, which is simply the negation of $p$. Then, we calculate the implication $p\Rightarrow q$.

Remember, an implication is only False when the hypothesis is True and the conclusion is False. This is the 'promise' rule—the only way to break a promise is to have the condition met but the action unfulfilled.

Next, we find the conjunction $(\sim p\wedge q)$, which is True only when both $\sim p$ and $q$ are True. Finally, we combine these using the OR operator $(\vee )$.

When we look at the final column for $S1$, we see that in the second row, the result is False. Because it is not True in every single case, $S1$ is not a tautology.

Deconstructing $S2$ (The Contradiction Hunt)

Now, let us turn our attention to $S2$:

This looks more intimidating, but we have already done the heavy lifting. We need the implication $q\Rightarrow p$, which is False only when $q$ is True and $p$ is False.

We then take this as our hypothesis and $(\sim p\wedge q)$ as our conclusion. We apply the implication rule again.

As we fill out the final column, we are looking for a contradiction—a column of all False values. However, in the third row, we find a True value. This single True value shatters the possibility of $S2$ being a contradiction.

The Elegance of Verification

We have systematically walked through every possibility. We found that $S1$ is not a tautology because it contains a False, and $S2$ is not a contradiction because it contains a True.

Therefore, neither statement is True according to the claims.

This problem teaches us that in logic, as in life, we must be methodical. We cannot guess; we must verify. I hope this journey has made the structure of logic feel less like a set of rules and more like a beautiful, predictable architecture.