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

In summary, the conversation discusses proving a statement by contradiction and determining whether finding a contradiction in one statement is sufficient or if it needs to be found in multiple statements. The conclusion is that the negation of "if P then (Q and R)" is "P and (not Q or not R)", and the contrapositive has the same truth value as the original statement.
  • #1
christoff
123
0
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)

?
 
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  • #2
Never mind. I figured it out.
 
  • #3
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).
 
  • #4
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)"
 
  • #5


I would say that finding a contradiction while assuming P and [(not Q1) and (not Q2)] is sufficient to prove the statement P implies (Q1 and Q2) by contradiction. This is because the assumption of P and [(not Q1) and (not Q2)] already covers the possibility of both P and (not Q1) and P and (not Q2) being true, and finding a contradiction in this case would prove that both Q1 and Q2 must be true for P to hold. However, if you wanted to be thorough, you could also try to find a contradiction in both P and (not Q1) and P and (not Q2) separately, to cover all possibilities. Ultimately, it depends on the level of rigor and detail required for the specific situation.
 

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

1. What is the concept of proof by contradiction?

Proof by contradiction is a method of mathematical proof in which a statement is proven to be true by assuming its opposite and showing that this assumption leads to a contradiction. This proves that the opposite of the statement must be false, therefore the original statement is true.

2. How does proof by contradiction apply to statements of the form P->(Q and R)?

In this case, proof by contradiction involves assuming that P is true, but Q or R (or both) are false. By showing that this assumption leads to a contradiction, we can prove that the original statement "P implies (Q and R)" must be true.

3. Is proof by contradiction the only way to prove statements of the form P->(Q and R)?

No, there are other methods of proof that can be used to prove statements of this form, such as direct proof or proof by contrapositive. However, proof by contradiction can be a useful tool in certain cases where other methods may be difficult to apply.

4. What are the steps involved in a proof by contradiction for P->(Q and R)?

The steps involved in a proof by contradiction for this statement would be:

  1. Assume P is true and Q or R (or both) are false.
  2. Show that this assumption leads to a contradiction.
  3. Therefore, the original statement "P implies (Q and R)" must be true.

5. Can proof by contradiction be used for any type of statement?

No, proof by contradiction is not applicable to all types of statements. It is most commonly used for statements involving logical implications, such as "if-then" statements. It may also be used for proving the existence of mathematical objects.

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