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

`\\eg \bigg( \\eg s \lor \big( \\eg r \land s \big) \bigg)`

where
`\\eg x` means "not x",
`\lor` means "or", and
`\land` means "and".

If r is false, then
`\\eg r` is true.

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

If s is false, then
`\\eg s` is true.

Then
`\\eg s \lor \big(\\eg 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

`\\eg\bigg(\\eg s \lor \big(\\eg r \impliedby s \big)\bigg)`

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