Question:
For any two statements \(p\) and \(q\), the statement \( \sim(p \vee q) \vee (\sim p \wedge q) \) is equivalent to
For any two statements \(p\) and \(q\), the statement \( \sim(p \vee q) \vee (\sim p \wedge q) \) is equivalent to
Show Hint
Always try factoring common logical terms to simplify expressions quickly.
Updated On: May 8, 2026
- \( \sim p \)
- \( p \)
- \( q \)
- \( \sim q \)
- \( p \vee q \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
Use logical identities:
• De Morgan's Law: \( \sim(p \vee q) = \sim p \wedge \sim q \)
• Distributive Law
• Absorption Law
Step 1: Apply De Morgan's Law
\[ \sim(p \vee q) = \sim p \wedge \sim q \] So expression becomes: \[ (\sim p \wedge \sim q) \vee (\sim p \wedge q) \]
Step 2: Factor common term
\[ = \sim p \wedge (\sim q \vee q) \]
Step 3: Use tautology
\[ \sim q \vee q = 1 \]
Step 4: Simplify
\[ = \sim p \wedge 1 = \sim p \]
Step 5: Final Answer
\[ \boxed{\sim p} \]
• De Morgan's Law: \( \sim(p \vee q) = \sim p \wedge \sim q \)
• Distributive Law
• Absorption Law
Step 1: Apply De Morgan's Law
\[ \sim(p \vee q) = \sim p \wedge \sim q \] So expression becomes: \[ (\sim p \wedge \sim q) \vee (\sim p \wedge q) \]
Step 2: Factor common term
\[ = \sim p \wedge (\sim q \vee q) \]
Step 3: Use tautology
\[ \sim q \vee q = 1 \]
Step 4: Simplify
\[ = \sim p \wedge 1 = \sim p \]
Step 5: Final Answer
\[ \boxed{\sim p} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM mathematical reasoning Questions
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
- Which one of the following is not a statement?
- If $p : 2$ plus $3$ is five and $q $ : Delhi is the capital of India < are two statements, then the statement "Delhi is the capital of India and it is not that $2$ plus $3$ is five" is
- The statement $p \rightarrow \left(\sim q\right)$ is equivalent to
- If $p :$ It is snowing, $q :$ I am cold, then the compound statement "It is snowing and it is not that I am cold" is given by
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions