Proof by contradiction for statement of the form P->(Q and R)

christoff

Say I have a statement like this:
P implies (Q1 and Q2).

If I wanted to prove this by contradiction, I would assume P and not(Q1 and Q2)=[(not Q1) or (not Q2)] both hold, and try to find a contradiction.

My question is... Am I done if I find a contradiction while assuming P and [(not Q1) and (not Q2)] ? Is this sufficient? Or do I need to find a contradiction in both the statements:
P and (not Q1),
P and (not Q2)

?

christoff

Never mind. I figured it out.

HallsofIvy

No. The negation of "Q and R" is "not Q or not R".
The negation of "if P then (Q and R)" is "If (not Q or not R) then not P".

(For those who read this thread and wondered).

jbriggs444

The contrapositive of "if P then (Q and R)" is "if (not Q or not R) then not P"

But the contrapositive has the same truth value as the original.

The negation of "if P then (Q and R)" is "P and (not Q or not R)"