Animated Solution for Mathematics - Basic Mathematics: Given the following two statements: $(s_1):(q\vee p)\to (p\leftrightarrow \sim q)$ is a tautology : $(s_2):\sim q\wedge (\sim p\leftrightarrow q)$ is a fallacy. Then :

Navigating the World of Logic

Welcome, future engineers! Today, we are stepping into the elegant, binary world of Mathematical Reasoning. Often, students view logic as a dry list of rules, but I want you to see it as the very foundation of the code that runs our digital world.

When we talk about tautologies and fallacies, we are not just solving a textbook problem; we are testing the structural integrity of logical statements. Let us embark on this journey to dissect $(s_1)$ and $(s_2)$ with precision and curiosity.

Building Our Truth Table

Before we dive into the complex expressions, we must establish our ground truth. We have two variables, $p$ and $q$. In the realm of classical logic, each can be either True ($T$) or False ($F$).

With two variables, we have $2^2=4$ possible scenarios. Imagine these as the four possible states of a system. We list them systematically: $(T,T)$, $(T,F)$, $(F,T)$, and $(F,F)$.

This is our canvas. Without this, we are lost in the fog of uncertainty. By laying out these four rows, we ensure that we have covered every possible reality of the universe for these variables.

The Gears of Logic

Now, we introduce the operators. Think of these as the gears in a machine. We have the negation $\sim$, the disjunction $\vee$ (OR), the conjunction $\wedge$ (AND), the implication $\to$, and the biconditional $\leftrightarrow$.

For $(s_1)$, we need to evaluate $(q\vee p)\to (p\leftrightarrow \sim q)$. Let us break this down.

The disjunction $q\vee p$ is the 'inclusive OR.' It is generous; it only demands that at least one input be True. It is False only when both $p$ and $q$ are False.

Next, the biconditional $p\leftrightarrow \sim q$ is the 'equality check.' It is True only when $p$ and $\sim q$ are identical. If you have $T\leftrightarrow F$, the machine breaks—it returns False. If you have $T\leftrightarrow T$, it returns True. This is the heart of the biconditional.

The Tautology Hunt

We are testing if $(s_1)$ is a tautology. A tautology is a statement that is always True, no matter what. It is a universal truth.

We look at the implication $(q\vee p)\to (p\leftrightarrow \sim q)$. An implication $A\to B$ is only False when $A$ is True and $B$ is False.

Let us check the first row where $p=T$ and $q=T$. Here, $q\vee p$ is $T\vee T=T$. Now, look at the second part: $p\leftrightarrow \sim q$. Since $p=T$ and $q=T$, then $\sim q=F$.

So, we have $T\leftrightarrow F$, which is False. Thus, we have an implication where the first part is True and the second part is False. The result is False! Because we found a case where the statement is False, $(s_1)$ fails the test of being a tautology. It is not a tautology.

The Fallacy Hunt

Now, let us turn our attention to $(s_2):\sim q\wedge (\sim p\leftrightarrow q)$. A fallacy is a statement that is always False. To prove it is not a fallacy, we only need to find one case where it is True.

Let us test the second row: $p=T,q=F$. Here, $\sim q$ is True. Now, look at the bracketed expression: $\sim p\leftrightarrow q$. Since $p=T$, $\sim p=F$. Since $q=F$, we have $F\leftrightarrow F$.

Since both sides match, this is True! Now, combine them with the AND operator: $\sim q\wedge (\sim p\leftrightarrow q)$ becomes $T\wedge T$, which is True. Since we found a case where the statement is True, it cannot be a fallacy. It is not a fallacy.

We have systematically dismantled both statements. We found that $(s_1)$ is not a tautology because it can be False, and $(s_2)$ is not a fallacy because it can be True.

Therefore, both statements are incorrect. This brings us to option (3).

Logic is not about memorizing tables; it is about the discipline of checking every possibility. When you approach these problems, do not rush. Build your table, respect the operators, and let the truth reveal itself. You have mastered the logic of this problem, and that is a skill that will serve you well beyond the exam hall!