Translate each English sentence into a logical expression using the propositiona

Question

Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English.

p: the applicant has written permission from his parents e: the applicant is at least 18 years old

s: the applicant is at least 16 years old

(a) The applicant has written permission from his parents and is at least 16 years old.

(b) The applicant has written permission from his parents or is at least 18 years old.

Explanation / Answer

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(a) The applicant has written permission from his parents and is at least 16 years old.

Solution a) logical expression: p ^ s

Then negate the entire logical expression using parentheses and the negation operation.
i.e.

=> ~(p ^ s)

Apply DeMorgan's law:

=> ~p v ~s

English sentence for the following logical expression :

=> The applicant not has written permission from his parents or is at most 15 years old.

(b) The applicant has written permission from his parents or is at least 18 years old.

Solution b) logical expression: p v e

Then negate the entire logical expression using parentheses and the negation operation.
i.e.

=> ~(p v e)

Apply DeMorgan's law:

=> ~p ^ ~e

English sentence for the following logical expression :

=> The applicant not has written permission from his parents or is at most 17 years old.

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