Question

Write the compound statement, “if $p$, then $q$ and if $q$, then $p$” in symbolic form.
A) $({\text{p}} \wedge {\text{q)}} \wedge {\text{(q}} \wedge {\text{p)}}$
B) $({\text{p}} \Rightarrow {\text{q)}} \vee {\text{(q}} \Rightarrow {\text{p)}}$
C) $({\text{q}} \Rightarrow {\text{p)}} \wedge {\text{(p}} \Rightarrow {\text{q)}}$
D) $({\text{p}} \wedge {\text{q)}} \vee {\text{(q}} \wedge {\text{p)}}$

Answer 1

First of all after reading the question carefully, we get a good idea about the symbols. We are going to reach the solution by trial and error method. Then by following the explanation of options given below, you can get the correct option with a clear explanation.

Complete step-by-step answer:
Let us check all the options
Option A:
The option is wrong. Because the symbol between the $p$ and $q$ is the symbol for “or”. As it is wrong for the first statement, there is no need to check further.

Option B:
The option is correct. Because the statement, “if $p$ then $q$” is represented by the symbol “$p \Rightarrow q$”. Then the statement is followed by “and” which is represented by “$\vee$”. The next statement is “if $q$ then $p$” which is represented by “$q \Rightarrow p$”. If then also can be said as implies. It can be explained as if $p$ is true, then $q$ must be true.

Option C:
The option is wrong. Because it is said that “$p$ implies $q$” but not “$q$ implies $p$” in the statement at the first. As it is wrong for the first statement, there is no need to check further.

Option D:
The option is wrong. The given symbolic form contains “or” symbols where it is not at all used in the given compound statement. As it is wrong for the first statement, there is no need to check further.

So, the correct answer is “Option B”.

Note: Mathematical Statement: A meaningful composition of words which can be considered either true or false is called a mathematical statement Do not get confused among the symbols. A compound statement consists of two or more statements that are separated by logical connectors.