Animated Solution for Mathematics - Sets and Relations: Which of the following is equivalent to the Boolean expression $p\wedge \sim q$ ?

We are tasked with finding an expression logically equivalent to $p\wedge \sim q$. In the realm of formal logic, this expression represents a specific state where the proposition $p$ is true and the proposition $q$ is false.

Our goal is to identify which logical structure maps perfectly onto this state.

The Implication Identity

To manipulate these expressions, we rely on the fundamental identity for material implication:

This identity is essential because it allows us to convert conditional statements into simpler disjunctions. Think of an implication as a promise; the only way to break the promise $p\to q$ is if the condition $p$ is met, but the result $q$ does not follow.

Consequently, the negation of an implication, $\sim (p\to q)$, describes the exact scenario where $p$ is true AND $q$ is false.

Applying De Morgan's Laws

Let us evaluate the expression $\sim (p\to q)$ to see if it matches our target. First, we substitute the implication identity into the parenthesis:

Now, we apply De Morgan's Law, which states that $\sim (A\vee B)\equiv \sim A\wedge \sim B$. This law acts as a distributive property for logic, where the negation operator flips the disjunction ($\vee$) into a conjunction ($\wedge$).

Applying this to our expression:

1. The negation of the first term, $\sim (\sim p)$, simplifies to $p$. 2. The operator $\vee$ is flipped to $\wedge$. 3. The negation of the second term, $q$, becomes $\sim q$.

Combining these steps, we arrive at:

Through the systematic application of logical laws, we have demonstrated that $\sim (p\to q)$ is equivalent to $p\wedge \sim q$.

This result highlights the beauty of Boolean algebra. By using the implication identity to eliminate conditional arrows and applying De Morgan's laws to distribute negations, complex logical structures can be reduced to their simplest, most elegant forms.