Question

Rewrite each of the following statement in the form “p if and only if q”
s: If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

Answer 1

Hint: We will break the given statement into two simple statements and denote them by ‘p’ and ‘q’. Substitute the values of ‘p’ and ‘q’ in the required form to get the final answer.

Complete step by step answer:
We are given the statement “If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.”
We can see this is a statement of the kind where the part after the word ‘and’ is just the reciprocal of the part before the word ‘and’.
i.e. ‘If a tumbler is half empty, then it is half full’ and ‘if a tumbler is half full, then it is half empty.’
\[\because \] ‘and’ is the logical operator joining two statements that are reciprocal of each other.
\[\therefore \]We can consider the one half of the complete sentence for the values of ‘p’ and ‘q’.
We can break the above given compound sentence into two simple statements as “A tumbler is half empty” and “A tumbler is half full”.
\[ \Rightarrow \]The statement “A tumbler is half empty” is p.
\[ \Rightarrow \]The statement “A tumbler is half full” is q.
We are required to rewrite the statement in the form “p if and only if q”.
Substitute the values of ‘p’ and ‘q’ in the required form:
\[ \Rightarrow \]“A tumbler is half empty if and only if it is half full.”
Another form of the required statement can be
\[ \Rightarrow \]“A tumbler is half full if and only if it is half empty.”

Note:
Students might get confused with the compound statement and take the whole part of the sentence before the logical operator ‘and’ and ‘p’ and after the operator as ‘q’ which is wrong. Keep in mind we check the statements joined by logical operators are reciprocal of each other if they can be written in the form: ‘if p then q’ and ‘if q then p’.