Animated Solution for Mathematics - Sets and Relations: The statement $(p\wedge (p\to q)\wedge (q\to r))\to r$ is :

Unlocking the Logic of Syllogism

Welcome, fellow traveler on the path of mathematical discovery. Today, we aren't just solving a problem; we are peeling back the curtain on how human thought itself is structured.

We are looking at the expression $(p\wedge (p\to q)\wedge (q\to r))\to r$. At first glance, it looks like a dense thicket of symbols, but I want you to see it as a beautiful, interconnected chain of reasoning.

This is the essence of deductive logic—the very foundation upon which all of science and mathematics is built.

Breaking the Implication Barrier

When we face a logical expression, our first instinct might be to panic at the sight of the implication arrow $\to$. But remember, the arrow is not a wall; it is a bridge.

We have a powerful tool in our arsenal: the equivalence $A\to B\equiv \sim A\vee B$. This identity allows us to translate the 'if-then' structure into the language of 'OR' and 'NOT.'

By applying this to our expression, we transform $(p\to q)$ into $(\sim p\vee q)$ and $(q\to r)$ into $(\sim q\vee r)$. Suddenly, the expression becomes:

We have moved from a complex conditional statement to a structure we can manipulate with the familiar rules of algebra.

The Art of Distribution

Now, let us look at the first part: $p\wedge (\sim p\vee q)$. Imagine you are distributing a number into a bracket; logic follows the same rhythm.

Using the Distributive Law, we get $(p\wedge \sim p)\vee (p\wedge q)$.

Here is where the magic happens. Look closely at $(p\wedge \sim p)$. This is the Law of Contradiction. A statement cannot be both true and false at the same time.

Therefore, $(p\wedge \sim p)$ is always False ($F$). Our expression simplifies beautifully to:

Since $F\vee X$ is just $X$, we are left with $(p\wedge q)\wedge (\sim q\vee r)\to r$. We are stripping away the noise, leaving only the signal.

The Final Simplification

We are almost there. Let us distribute $(p\wedge q)$ across the second bracket:

Again, we encounter the Law of Contradiction: $q\wedge \sim q$ is $F$. This collapses the first term entirely, leaving us with $(p\wedge q\wedge r)\to r$.

Now, we apply our implication rule one last time to the entire expression: $\sim (p\wedge q\wedge r)\vee r$. Using De Morgan's Law, we expand the negation:

The Law of Excluded Middle

Look at the final grouping: $\sim p\vee \sim q\vee (\sim r\vee r)$. The term $(\sim r\vee r)$ is the Law of Excluded Middle.

It is the bedrock of logic—a statement is either true or it is not. Thus, $(\sim r\vee r)$ is always True ($T$).

When you OR anything with True, the result is always True. The entire expression collapses into $T$.

We have proven that this statement is a tautology. It is a statement that is true by its very structure, independent of the values of $p,q,$ or $r$.

You have just navigated the architecture of logical certainty. Take a moment to appreciate the elegance of this result—the way the contradictions cancelled out and the excluded middle brought us to the final, undeniable truth.