Question

Please look at the included picture, show me your work and please include a good explanation. It would really help me out here.

Answer 1

These are indeed equivalent, and this identity is one of DeMorgan's laws.

The 6th column is the negation of the 5th column. For example, the first row says

not p or q

is true (T), so the negation would be false (F). The 5th column reads {T, F, T, T}, so the next column should be {F, T, F, F}.

Then in the 7th/last column, you are checking the truth value of the statement

p and not q

For example, in the first row, both p and q are true (T). This means (not q) is false, so (p and not q) is false (F). The last column should end up reading {F, T, F, F}, same as the previous column.