The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:
The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:
Show Hint
To find the negation of logical expressions, apply De Morgan's laws carefully, then simplify using logical identities and set theory concepts if needed.
- \( p \land (\sim q) \)
- \( p \land q \)
- \( p \land (q \land (\sim p)) \)
- \( p \lor (q \lor (\sim p)) \)
The Correct Option is B
Solution and Explanation
We start with the given expression:
\[ (p \land (\sim q)) \lor (\sim p). \]
Apply the negation:
\[ \sim \big((p \land (\sim q)) \lor (\sim p)\big). \]
Using De Morgan's laws:
\[ \sim (A \lor B) = (\sim A) \land (\sim B). \]
Here, \( A = (p \land (\sim q)) \) and \( B = (\sim p) \):
\[ \sim \big((p \land (\sim q)) \lor (\sim p)\big) = (\sim (p \land (\sim q))) \land (\sim (\sim p)). \]
Simplify each term:
- For \( \sim (p \land (\sim q)) \):
Using De Morgan's laws:
\[ \sim (p \land (\sim q)) = (\sim p) \lor q. \] - For \( \sim (\sim p) \):
\[ \sim (\sim p) = p. \]
Thus, the expression becomes:
\[ ((\sim p) \lor q) \land p. \]
Distribute \( p \):
\[ ((\sim p) \land p) \lor (q \land p). \]
Since \( (\sim p) \land p = \text{False} \):
\[ \text{False} \lor (q \land p) = (q \land p). \]
Hence, the negation simplifies to:
\[ p \land q. \]
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Concepts Used:
Logic Gates
AND Gate
It is the gate, where a circuit performs an AND operation. It has n number of input where (n >= 2) and one output.
OR Gate
It is the gate, where a circuit performs an OR operation. It has n number of input where (n >= 2) and one output.
NOT Gate
An inverter is also called NOT Gate. It has one input and one output where the input is A and the output is Y.
NAND Gate
A NAND operation is also called a NOT-AND operation. It has n number of input where (n >= 2) and one output.
NOR Gate
A NOR operation is also called a NOT-OR operation. It has n number of input where (n >= 2) and one output.
XOR Gate
XOR or Ex-OR gate is a specific type of gate that can be used in the half adder, full adder, and subtractor.
XNOR Gate
XNOR gate is a specific type of gate, which can be used in the half adder, full adder, and subtractor. The exclusive-NOR gate is flattened as an EX-NOR gate or sometimes as an X-NOR gate. It has n number of input (n >= 2) and one output.