Question

The dual of the statement “Ram and Laxman are brothers “ is,\[\]
A. Ram or Laxman are brothers. \[\]
B. Ram and Laxman are not brothers . \[\]
C. Ram and Laxman are brothers . \[\]
D. Ram and Laxman are brothers .\[\]

Answer 1

Hint: We denote the given statement “Ram and Laxman are brothers ” as $p$. We know that the dual of $p$ is the negation of $p$ that is $\tilde{\ }p.$ We write the $\tilde{\ }p.$ statement in a sentence and check the option to match.\[\]

Complete step by step answer:
We know that declarative sentences can be logically investigated when we can assign a truth value T or F but not both unequivocally and assertively. We then call the declarative sentence a logical statement or mathematical statement. It is denoted as $p,q,r...$\[\]
When we want to join two prime statements we use logical connectives and get a composite statement. When we compose a statement the truth values operate within themselves. We know that there are 3 type of binary operations conjunction $\left( \wedge \right)$, disjunction $\left( \vee \right)$, implication $\left( \to \right)$ and one unitary operation negation$\left( \tilde{\ } \right)$. \[\]
We also know that the dual of statement is obtained as by replacing the conjunction $\left( \wedge \right)$ with disjunction $\left( \vee \right)$ and vice –versa. We also find dual of the statement $p$ as $\tilde{\ }p.$ \[\]
The given statement is “Ram and Laxman are brothers”. We assume the truth value with respect to mythological knowledge as conventional and we assign a truth value T onto the statement. We see the word “and” here which is primarily used as logical connective during conjunction operation in the composition of the statement but here we can see that there is only one statement and it is without any prime statement. \[\]
So let us denote the given statement as $p$. The dual of $p$ is $\tilde{\ }p.$ We can write the negation of $p$ as “Ram and Laxman are not brothers.” We check the options and find the statement in option B. So the correct choice is option B. \[\]

Note:
We note that if the statement $p$ has a truth value T or F then the negation of $p$ is denoted as $\tilde{\ }p$ and has truth value F or T respectively. So the truth value of the negation of the given statement here is F. We can also find the dual of $p\wedge q$ as $p\vee q$.