Question

Consider the following two statements:
P: If $7$ is an odd number, then $7$ is divisible by $2$.
Q: If $7$ is prime number, then $7$ is an odd number.
If ${V_1}$ is the truth value of the contrapositive of P and ${V_2}$ is the truth value of contrapositive of Q, then the ordered pair $({V_1},{V_2})$ equals:
$A)\,(F,F) \\\ B)\,(T,T) \\\ C)\,(T,F) \\\ D)\,(F,T) \\\$

Answer 1

At first find out the contrapositive of the given statements. Then find the truth value of these statements. Hence you will get the value of $({V_1},{V_2})$.

Complete step-by-step answer:
We know we have to find the contrapositive of the given statements P and Q. but what does contrapositive mean? Just the opposite of a statement ‘is’ is changed to ‘is not’ in the statements.
For example: $7$ is prime number
It’s contrapositive is $7$ is not a prime number
Contrapositive statement of P:
If $7$ is not an odd number, then $7$ is not divisible by $2$.
Contrapositive statement of Q:
If $7$ is not a prime number, then $7$ is not an odd number.
Now we will find their truth values. Truth values are determined by the following table where ${P_1}\,\&\, {P_2}$ are the first and second halves of the student:

Hence for statement P, not ${P_2}$ is true and not ${P_1}$ is false.
Not ${P_2}$:$7$ is not divisible by $2$
Not ${P_1}$: $7$ is not an odd number
Hence truth table of P is False
Similarly for statement Q, not ${Q_2}$ is false and not ${Q_1}$ is false
Hence Q is True
$\Rightarrow {V_1} = F\,\,\,\,\& \,\,\,{V_2} = T$
Therefore ordered pair is $(F,T)$

So, the correct answer is “Option D”.

Note: A truth table is a mathematical table used to determine if a compound statement is true or false.Remembering the table is very much necessary as it is the most important part. If you don’t remember, you won’t be able to find the truth value and hence cannot solve the question.Also, ${P_1} \Rightarrow {P_2}$ is same as $not{P_1} \Rightarrow \,not{P_2}$