Question

p: He is hard working.
q: He will win.
The symbolic form of “If he will not win then he is not hard working”, is
(a) \[p\Rightarrow q\]
(b) \[\left( \sim p \right)\Rightarrow \left( \sim q \right)\]
(c) \[\left( \sim q \right)\Rightarrow \left( \sim p \right)\]
(d) \[\left( \sim q \right)\Rightarrow p\]

Answer 1

Hint: We start solving the problem by recalling the definition of negation of statement as the statement that is made opposite by adding not to the given statement. We then write the negation for the statements p and q. We then make a replacement of the statements with the symbols p or q or ~p or ~q that were present in “If he will not win then he is not hard-working”. We then recall the if-then connective and make use of it to get the required answer.

Complete step-by-step solution
According to the problem, we are given two statements represented as p: He is hard working and q: He will win. We need to find the symbolic form of the statement “If he will not win then he is not hard-working”.
Let us write the negation of the given statements p and q.
We know that negation of a statement is defined as the statement that is made opposite by adding not to the given statement.
So, we get ~p: He is not hardworking and ~q: He will now win.
Now, let us write the symbolic form for the statement “If he will not win then he is not hard-working”.
So, we have “If ~q then ~p”
We can see that the two statements of ~q and ~p are connected by if-then (implication) connective. We know that the implication of the two statements is denoted by $\Rightarrow $.
So, the symbolic form of the statement “If he will not win then he is not hard working” will be $\left( \sim q \right)\Rightarrow \left( \sim p \right)$.
$\therefore$ The correct option for the given problem is (c).

Note: Whenever we get this type of problem involving not in the final statements, we try to make use of negation property to reduce the calculation time. We should not confuse one connective with other connectives while solving this problem. Similarly, we can make use of the property $\left( p\Rightarrow q \right)=\left( \left( \sim q \right)\Rightarrow \left( \sim p \right) \right)$ to find the symbolic form of the statement “If he is hardworking then he will win”.