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Chapter 20: Problem 1916

\(\sim(p V \sim q) V(\sim p \wedge q)=\ldots \ldots . .\) (a) \(\mathrm{p}\) (b) \(\sim \mathrm{p}\) (c) \(q\) (d) \(\sim \mathrm{q}\)

Short Answer

Expert verified
The given logical expression \(\sim(p V \sim q) V(\sim p \wedge q)\) is equivalent to the logical expression \(\sim q\) (option (d)).

Step by step solution

01

Apply De Morgan's Law

De Morgan's Law states that \(\sim(A \wedge B) \equiv \sim A V \sim B\) and \(\sim(A V B) \equiv \sim A \wedge \sim B\). Applying De Morgan's law to the given expression, we get: \((\sim p \wedge q) V (\sim p \wedge q)\)
02

Distribute \(\wedge\) over \(V\)

According to the distributive law, \(A \wedge (B V C) \equiv (A \wedge B) V (A \wedge C)\). Applying distributive law to the expression obtained in step 1, we get: \((\sim p \wedge q) V (\sim p \wedge q)\) Notice that the expression did not change. This is because this expression is already in its simplest form.
03

Evaluate the expression

We are now supposed to match our simplified expression with the given options. Evaluating our simplified expression, we find that it matches with the logical expression for option (d). So, the given logical expression \(\sim(p V \sim q) V(\sim p \wedge q)\) is equivalent to the logical expression \(\sim q (option\,(d))\).

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Most popular questions from this chapter

If both \(\mathrm{p}\) and \(\mathrm{q}\) are false then (a) \(\mathrm{p} \Lambda \mathrm{q}\) is true (b) \(\mathrm{p} \mathrm{V} \mathrm{q}\) is false (c) \(p \Rightarrow q\) is false (d) \((\sim p) V q\) is false

\(\mathrm{p} \Rightarrow \mathrm{q} \mathrm{V} \mathrm{r}\) is false then the true values of \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) are respectively. (a) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (d) \(\mathrm{F}, \mathrm{F}, \mathrm{F}\)

Negation of for all \(\mathrm{x}, \mathrm{p}^{\prime}\) is \(\ldots \ldots \ldots\) (a) there exists \(\mathrm{x}, \sim \mathrm{p}\) (b) for all \(\mathrm{x} ; \sim \mathrm{p}\) (c) \(\sim \mathrm{p}\) (d) \(\mathrm{p}\)

\(p \Rightarrow(q \Rightarrow p) \Rightarrow r\) is (a) Contradiction (b) tautology (c) Neither contradiction Nor tautology (d) Both contradiction \& tautology

Which one of the following is false (a) p \(\Lambda(\sim \mathrm{p})\) is a contradiction (b) \((\mathrm{p} \Rightarrow \mathrm{q}) \Leftrightarrow(\sim \mathrm{q} \Rightarrow \sim \mathrm{p})\) is a contradiction (c) \(\sim(\sim p) \Leftrightarrow p\) is a tautology (d) \(p \mathrm{~V}(\sim \mathrm{p})\) is a tautology
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