logical deduction question

Question

For example:

S ::= “Sales will go up.” E ::= “Expenses will go up.”

Express the following: Both sales and expenses will fall.

Why is it ¬ (S ∧ E) instead of ¬S ∧ ¬E ?

Answer 1

For starters, $\neg S:=$"sales will NOT go up" which is clearly different from saying that sales will fall. Basically, logical opposite versus polar opposite.

"It is not true that both sales and expenses will go up" can be trivially translated into symbols by $\neg (S \land E)$ which is logically equivalent to $\neg S \lor \neg E$.

Perhaps consider editing your question because there seems to be some issues that probably only you can rectify.

Answer 2

Your understanding is a bit flawed. The expression $\neg (S \land T)$ means "Both S and T are not true." (i.e. at least one of them is false.)

$\neg S \land \neg T$ means "Both S and T are false."