Question

Let s represent a false statement and let r represent a false statement. Find the truth value of the following statement.
~[~s v (~ r < s ) ]
Is the statement true or false?
True
False

Answer 1

I'll interpret the given statement as

$\neg \bigg( \neg s \lor \big( \neg r \land s \big) \bigg)$

where $\neg x$ means "not x", $\lor$ means "or", and $\land$ means "and".

If r is false, then $\neg r$ is true.

s is given to be false, so $\neg r\land s$ (basically "true and false") is false.

If s is false, then $\neg s$ is true.

Then $\neg s \lor \big(\neg r \land s\big)$ (i.e. "true or false") is true.

Take the negation of that and you end up with a false statement.

If you intended "~r < s" to mean something like "not r is implied by s", so the original statement is actually

$\neg\bigg(\neg s \lor \big(\neg r \impliedby s \big)\bigg)$

then $s\implies \neg r$ is true because s is false. Then $\neg s \lor \big(\neg r \impliedby s\big)$ is still true, so the statement still ends up being false.