Implication Operator

Question

I think I fail to understand why I might be wrong in the way I translate the following formulas in English.

D stand for 'I will eat'

R stand for 'I will drink'

1. ¬R -> ¬D

My answer: I will not eat if I will not drink.

Because of order of precedence and the not operator goes first:

Could be: I will not eat, I will not drink therefore.

2. ¬(D ∧¬R )

My answer: I will not eat and I will drink.

I think: The operation in the parenthesis evaluates to false.

Could it be: I will eat and I will not drink. ?

thank you for looking at the problem.

Update:

Possible answer for the second question:

I will drink and I will eat.

logic: evaluation in the () is false

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Second update

As per comment I am still missing out and I should read on De Morgan's laws not() => true

If i remove the parentheses I would have:

Possible answer:

I will not eat and I will not drink.

Answer 1

• D stand for 'I will eat'
• R stand for 'I will drink' \begin{equation}\tag{1} \lnot R\rightarrow \lnot D \end{equation} \begin{equation}\tag{2} \lnot(D\land\lnot R) \end{equation}

Your first answers are correct, although I had to read your second attempt twice before I understood the meaning. I would personally write $(1)$ as "If I will not drink, then I will not eat".

In general, when it's not that case that two things ($D$ and $\lnot R$) are true, at least one of them is false. This means the English translation of $(2)$ would be "Either I will drink, or I will not eat, or both".