Question

If c denotes the contradiction then dual of the compound statement $\sim p\wedge \left( q\vee c \right)$ is :
A. $\sim p\vee \left( q\wedge t \right)$
B. $\sim p\wedge \left( q\vee t \right)$
C. $p\wedge \left( \sim q\vee t \right)$
D. $\sim p\vee \left( q\wedge c \right)$

Answer 1

Hint: We can clearly tell that this question is from mathematical logic. We should be aware of what each symbol and letter means in this chapter. The symbol $\sim $ stands for negation. The symbol $\wedge $ stands for intersection and the symbol $\vee $ stands for union. In the definition of duality of the given compound statement, the symbols change . The negation symbol does not change. The intersection symbol changes to union symbol. $t$ stands for tautology . So the contradiction symbol $c$ changes to $t$ .

Complete step-by-step solution:
The given compound statement is $\sim p\wedge \left( q\vee c \right)$.
This is the compound statement since it is a combination of symbols and letters.
We have to find out the dual of this compound statement.
We are already see what changes are made when we convert a compound statement into it’s dual compound statement.
Let us first change the intersection symbol $\left( \wedge \right)$ to union symbol $\left( \vee \right)$.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \sim p\wedge \left( q\vee c \right) \\
 & \Rightarrow \sim p\vee \left( q\vee c \right) \\
\end{align}$
Let us first change the union symbol $\left( \vee \right)$ to intersection symbol $\left( \wedge \right)$.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \sim p\wedge \left( q\vee c \right) \\
 & \Rightarrow \sim p\vee \left( q\vee c \right) \\
 & \Rightarrow \sim p\vee \left( q\wedge c \right) \\
\end{align}$
We should also change the contradiction symbol $\left( c \right)$ to tautology symbol $\left( t \right)$ .
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \sim p\wedge \left( q\vee c \right) \\
 & \Rightarrow \sim p\vee \left( q\vee c \right) \\
 & \Rightarrow \sim p\vee \left( q\wedge c \right) \\
 & \Rightarrow \sim p\vee \left( q\wedge t \right) \\
\end{align}$
$\therefore $ If $c$denotes the contradiction then dual of the compound statement $\sim p\wedge \left( q\vee c \right)$ is $\sim p\vee \left( q\wedge t \right)$.

Note: The negation symbol doesn’t change. We should be aware of all the symbols and their definitions. We should also know how to draw truth tables. This chapter is quite easy . This chapter is quite scoring too. So, if we can practice more, we can do it quickly. We can also look at previous years problems so as to get an idea of how the questions might be. We should not get confused between the symbols especially between the intersection symbol and the union symbol.