Animated Solution for Mathematics - Basic Mathematics: The logical statement $(p\to q)\wedge (q\to \sim p)$ is equivalent to

Unlocking the Truth Table

Welcome, future engineers! Today, we are not just solving a problem; we are peeling back the layers of logical reasoning.

The expression $(p\to q)\wedge (q\to \sim p)$ might look like a jumble of symbols, but it is actually a precise, elegant structure waiting to be decoded. Imagine you are a detective, and these variables $p$ and $q$ are suspects. We need to find out when their combined testimony holds up using the Truth Table.

The Foundation

We start with the base variables $p$ and $q$. Since each can be either True ($T$) or False ($F$), we have $2^2=4$ possible scenarios.

We list them systematically: $(T,T),(T,F),(F,T),$ and $(F,F)$. This is our map; without this, we are lost in the woods.

Deconstructing the Implication

Now, let's tackle the first bracket: $p\to q$. Remember the golden rule of implication: it is only False when a True premise leads to a False conclusion.

In our table, this happens only in the second row where $p=T$ and $q=F$. Everywhere else, the implication is True.

Next, we prepare for the second bracket by finding $\sim p$. This is simply the inversion of the $p$ column: If $p$ is $T$, $\sim p$ is $F$. If $p$ is $F$, $\sim p$ is $T$.

Now, we evaluate $q\to \sim p$. Applying the same implication rule, we look at the $q$ column and the $\sim p$ column. The only row that yields a False is the first row, where $q$ is True and $\sim p$ is False.

The Final Conjunction

We have our two components: $(p\to q)$ and $(q\to \sim p)$. Now, we combine them with the conjunction operator $\wedge$.

The rule for $\wedge$ is strict: it is only True if both components are True. Let's look at our results: Row 1: $T\wedge F=F$ Row 2: $F\wedge T=F$ Row 3: $T\wedge T=T$ Row 4: $T\wedge T=T$

Our final column values are $F,F,T,T$.

Look at the final column. It reads $F,F,T,T$. Now, look back at the column we generated for $\sim p$. It is also $F,F,T,T$.

They are identical! This is the beauty of mathematical logic. We have proven that:

You have successfully navigated the logic maze. Keep this methodical approach in your toolkit, and no logical statement will ever intimidate you again.