Animated Solution for Mathematics - Sets and Relations: Let $r\in \{p,q,\sim p,\sim q\}$ be such that the logical statement $r\vee (\sim p)\Rightarrow (p\wedge q)\vee r$ is a tautology. Then 'r' is equal to :

Mastering Logical Reasoning

Welcome, future engineers! Today, we are not just solving a problem; we are exploring the very bedrock of mathematical thought.

Logic is the language of the universe, and tautologies are its absolute truths. When we are asked to find an $r$ that makes a statement a tautology, we are essentially looking for the missing piece of a puzzle that, when placed, makes the entire structure unshakable.

The Implication Identity

We are given the statement $r\vee (\sim p)\Rightarrow (p\wedge q)\vee r$. At first glance, the implication arrow $\Rightarrow$ can feel like a barrier.

It is not a standard algebraic operator, which makes it tricky to manipulate. But here is the secret: every implication can be translated using the identity:

This is our bridge. By converting the implication into an 'or' statement, we unlock the ability to use the full power of Boolean algebra—distributive laws, absorption, and the law of excluded middle.

Testing the Waters

We have four candidates for $r$: $p$, $q$, $\sim p$, and $\sim q$. We must test them.

Let us start with $r=p$. Substituting this into our expression, we get:

The left side is a classic: $p\vee \sim p$ is always $T$. The right side, using the Absorption Law, simplifies to $p$. So, we are left with $T\Rightarrow p$, which simplifies to $p$. Since $p$ is not always true, this is not a tautology.

Testing $r=q$ leads us to a similar dead end. The expression simplifies to $p\vee q$, which depends on the truth values of $p$ and $q$. It is not a universal truth.

The Elegance of $\sim p$

Now, let us test the third option: $r=\sim p$. This is where the magic happens. Substitute it into the original statement:

First, look at the left side. By the Idempotent Law, $\sim p\vee \sim p$ simply collapses into $\sim p$.

Now, look at the right side: $(p\wedge q)\vee \sim p$. This is a beautiful application of the Distributive Law. We distribute the $\vee \sim p$ across the conjunction:

Since $p\vee \sim p$ is $T$, the entire right side simplifies to $q\vee \sim p$.

Now, bring it all together. Our statement is now $\sim p\Rightarrow (q\vee \sim p)$. Applying our implication identity one last time, we get:

The double negation $\sim (\sim p)$ becomes $p$. We are left with $p\vee q\vee \sim p$.

Look at what we have created: $p\vee q\vee \sim p$. By the Associative Law, we can rearrange this as $(p\vee \sim p)\vee q$.

We know that $p\vee \sim p$ is $T$. Thus, we have $T\vee q$. In the world of logic, $T\vee q$ is always $T$.

We have done it! We have transformed a complex, intimidating expression into a simple, undeniable truth. The missing piece was $\sim p$ all along.