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:
\[ p \land q. \]
Top Questions on Logic gates
The output (Y) of the given logic implementation is similar to the output of an/a …………. gate.
- NEET (UG) - 2025
- Physics
- Logic gates
- How many AND gates are required to realize \( Y = CD + EF + G \)?
The logic gate equivalent to the circuit given in the figure is
- AP EAPCET - 2025
- Physics
- Logic gates
The logic gate equivalent to the combination of logic gates shown in the figure is
- AP EAPCET - 2025
- Physics
- Logic gates
- Which logic gate is represented by the following combination of logic gates?
- WBJEE - 2025
- Physics
- Logic gates
Questions Asked in JEE Main exam
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
- Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
- JEE Main - 2025
- Vectors
- Three infinitely long wires with linear charge density \( \lambda \) are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?
- JEE Main - 2025
- electrostatic potential and capacitance
- The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is:
- JEE Main - 2025
- Calculus
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
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.