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

Welcome, future engineers! Today, we are stepping into the elegant, binary world of mathematical reasoning. Often, students view logic problems as tedious chores, but I want you to see them as the bedrock of computer science and advanced mathematics.

We are going to dissect the expression $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ and uncover its hidden simplicity. Imagine you are an architect building a bridge; you need to ensure that every possible condition is accounted for. That is exactly what we do with a truth table.

The Landscape of Possibilities

We start with two variables, $p$ and $q$. Since each can be either True ($T$) or False ($F$), we have $2^2=4$ possible combinations. Think of these as the four corners of our logical universe:

1. $p=T,q=T$ 2. $p=T,q=F$ 3. $p=F,q=T$ 4. $p=F,q=F$

By laying these out, we create the foundation for our analysis. Never skip this step; it is the map that prevents you from getting lost in the abstraction.

The Implication—The Promise

Now, let us tackle the implication operator, $\Rightarrow$. This is the most misunderstood operator in JEE logic. Remember the golden rule: an implication $A\Rightarrow B$ is only False when the premise $A$ is True and the conclusion $B$ is False. In every other case, it is True.

For the first part, $(p\Rightarrow q)$, we look at our table. In the second row, $p$ is True and $q$ is False, so the implication is False. Everywhere else, it is True.

Now, for the second part, $(q\Rightarrow \sim p)$, we treat $q$ as the premise and $\sim p$ as the conclusion. We look for the row where $q$ is True and $\sim p$ is False. That happens in the first row. So, the first row is False, and the rest are True.

The Intersection—The AND Operator

We have our two implication columns. Now, we use the conjunction operator, $\wedge$ (AND). The rule here is strict: the result is True only if both inputs are True. If even one is False, the whole thing collapses into False.

Let us look at our rows:

- Row 1: $T\wedge F=F$ - Row 2: $F\wedge T=F$ - Row 3: $T\wedge T=T$ - Row 4: $T\wedge T=T$

The Revelation

Look at the final column we just generated: $(F,F,T,T)$. Now, compare this to the column for $\sim p$.

If $p$ is $(T,T,F,F)$, then $\sim p$ is $(F,F,T,T)$. They are identical!

We have successfully reduced a complex compound statement into a simple negation. This is the beauty of logic—finding the hidden simplicity beneath the complexity. Keep practicing these tables, and you will find that even the most intimidating logical expressions become clear and manageable. You have got this!