Question

Rewrite each of the following statements in the form of “p if and only if q”
(i) P: If you watch television then your mind is free and if your mind is free then you watch television.
(ii) q: For you to get an A grade it is necessary and sufficient that you do all the homework regularly.
(iii) r: If a quadrilateral is equiangular then it is a rectangle and if a quadrilateral is a rectangle then it is equiangular.

Answer 1

Hint: Here we will proceed by separating the statements into two different statements as p and q. further rewrite the separated statements as “p if and only if q” to get the required answer of the given problem.

Complete step-by-step answer:
Here we have to rewrite the statements as of the form of “p if and only if q”.

(i) P: If you watch television then your mind is free and if your mind is free then you watch television.
Now we will separate the statements as p and q. so, we have p as “You watch television” and q as “Your mind is free ''.
Thus, the required statement is “You watch television if and only if our mind is free”
Also, we can take the statements as p “Your mind is free” and q “You watch television”
Thus, the required statement is “Your mind is free if and only if you watch television”

(ii) q: For you to get an A grade it is necessary and sufficient that you do all the homework regularly.
Now we will separate the statements as p and q. so, we have p as “You get an A grade” and q as “You do all the homework regularly”.
Thus, the required statement is “You get an A grade if and only if you do all the homework regularly”.

(iii) r: If a quadrilateral is equiangular then it is a rectangle and if a quadrilateral is a rectangle then it is equiangular.
Now we will separate the statements as p and q. so, we have p as “A quadrilateral is equilateral '' and q as “A quadrilateral is rectangle”.
Thus, the required statement is “A quadrilateral is equiangular if and only if it is a rectangle”
Also, we can take the statements as p “A quadrilateral is rectangle” and q “A quadrilateral is equiangular”.
Thus, the required statement is “A quadrilateral is rectangle if and only if it is an equiangular”

Note: Always remember that A quadrilateral is rectangle if and only if it is an equiangular or A quadrilateral is equiangular if and only if it is a rectangle. For statements (i) and (ii) we can rewrite them as two different required statements as they are interlinked to each other.