Question

De Morgan's Laws Using De Morgan's Laws, rewrite the following sentences in English. a) You cannot be late and you cannot smoke. b) It is not the case that you can take an annuity and you can take a lump sum. To solve this exercise, it is useful to first translate the sentence from English into symbolic form, apply the appropriate De Morgan's law, and then translate the sentence back in English

Answer 1

Rewritten Sentence: You cannot take both an annuity and a lump sum.

Step-by-step explanation:

De Morgan's Laws are used to manipulate logical expressions involving negation (NOT) and conjunction (AND) or disjunction (OR). Let's rewrite the given sentences using De Morgan's Laws:

a) Original Sentence: You cannot be late and you cannot smoke.

First, let's represent the negations:

You can be on time (NOT late).

You do not smoke (NOT smoke).

Now, using De Morgan's Law for conjunction (AND):

It is not the case that (You can be on time AND You do not smoke).

Rewritten Sentence: You must be on time or you must not smoke.

b) Original Sentence: It is not the case that you can take an annuity and you can take a lump sum.

First, let's represent the negations:

You cannot take an annuity (NOT take an annuity).

You cannot take a lump sum (NOT take a lump sum).

Now, using De Morgan's Law for conjunction (AND):

It is not the case that (You can take an annuity AND You can take a lump sum).

Rewritten Sentence: You cannot take both an annuity and a lump sum.