Using laws of logic prove that following are equivalenti)~ ( ) and ~ p q p q → 

To prove the equivalence $\sim (p \land q)$ and $\sim p \lor \sim q$ using laws of logic, we can follow these steps:

1. Break Down the Problem

We need to prove that: $\sim (p \land q) \equiv \sim p \lor \sim q$ This is known as De Morgan's Law.

2. Relevant Concepts

We will use the following logical equivalences:

• De Morgan's Laws:
  1. $\sim (p \land q) \equiv \sim p \lor \sim q$
  2. $\sim (p \lor q) \equiv \sim p \land \sim q$

3. Analysis and Detail

Let's start with the left side of the equivalence:

1. Begin with $\sim (p \land q)$.
2. According to De Morgan's Law, we can rewrite this as: $\sim (p \land q) \equiv \sim p \lor \sim q$

4. Verify and Summarize

The transformation holds according to the definitions provided by De Morgan's Laws. Thus, we have shown that: $\sim (p \land q) \equiv \sim p \lor \sim q$ This concludes the proof.

Final Answer

Therefore, the two statements are equivalent: $\sim (p \land q) \equiv \sim p \lor \sim q$