Animated Solution for Mathematics - Sets and Relations: The negation of $(p\wedge (\sim q))\vee (\sim p)$ is equivalent to

Logic is the architecture of thought. When we look at an expression like $(p\wedge (\sim q))\vee (\sim p)$, it might look like a jumble of characters, but it is a map. Let us embark on this journey to find its negation.

Imagine you have two circles, $P$ and $Q$, on a whiteboard. The first part of our expression, $(p\wedge (\sim q))$, represents the region inside circle $P$ but strictly outside circle $Q$. Think of it as the 'P-only' zone.

The second part, $(\sim p)$, represents everything in the universe that is not inside circle $P$. If you combine these two regions using the OR operator ($\vee$), you are essentially shading almost the entire board. The only piece left unshaded is the intersection of $P$ and $Q$.

While the Venn diagram gives us the 'what,' algebra gives us the 'why.' We apply the Distributive Law to the expression $(p\wedge (\sim q))\vee (\sim p)$.

Distributing the $(\vee (\sim p))$ across the terms, we get:

Look at the first bracket: $(p\vee (\sim p))$. This is a Tautology, as a statement must either be true or false. Therefore, this bracket is always True ($T$).

Our expression now simplifies to $T\wedge ((\sim q)\vee (\sim p))$. In logic, any statement $X$ ANDed with True is simply $X$ itself. The tautology vanishes, leaving us with:

The question asks for the negation of the original statement. We apply De Morgan's Law, which states that $(\sim p)\vee (\sim q)$ is equivalent to $\sim (p\wedge q)$.

To find the negation of our simplified expression, we calculate:

The double negation cancels out, leaving us with the elegant, simple result: