The Boolean expression ~(p q) ( ~ p q) is equivalent to:

Engineering

Mathematics

Logical Operator Reasoning AND and OR

Question

The Boolean expression ~(p $\lor$ q) $\lor$ ( ~ p $\land$ q) is equivalent to:

~ p

~ q

Understanding the Boolean Expression

We are given the expression: $\sim (p \lor q) \lor (\sim p \land q)$

We need to simplify this expression and find which of the given options it is equivalent to.

Step 1: Apply De Morgan's Law

First, simplify $\sim (p \lor q)$ using De Morgan's Law:

$\sim (p \lor q) \equiv \sim p \land \sim q$

So the expression becomes: $(\sim p \land \sim q) \lor (\sim p \land q)$

Step 2: Factor Out Common Term

Notice that $\sim p$ is common in both parts of the OR expression:

$\sim p \land \sim q \lor \sim p \land q \equiv \sim p \land (\sim q \lor q)$

Step 3: Apply Complement Law

We know that $\sim q \lor q \equiv 1$ (always true):

So the expression simplifies to: $\sim p \land 1 \equiv \sim p$

Final Answer

The expression simplifies to $\sim p$ , which is equivalent to the first option: ~p

Key Formulae

De Morgan's Laws:

$\sim (p \land q) \equiv \sim p \lor \sim q$

$\sim (p \lor q) \equiv \sim p \land \sim q$

Complement Law: $p \lor \sim p \equiv 1$

Identity Law: $p \land 1 \equiv p$

Detailed Solution

Understanding the Boolean Expression

Step 1: Apply De Morgan's Law

Step 2: Factor Out Common Term

Step 3: Apply Complement Law

Final Answer

Related Topics

Key Formulae

Detailed Solution

Understanding the Boolean Expression

Step 1: Apply De Morgan's Law

Step 2: Factor Out Common Term

Step 3: Apply Complement Law

Final Answer

Related Topics

Key Formulae

Student who searched for similar questions

Questions 1

Questions 2

Questions 3

Questions 4

Questions 5