Proof By Contradiction Worksheet
ADVERTISEMENT
1
2
3Proof by Contradiction –MATH 3000
Kawai
p
(#1) How do we prove that
2 is irrational?
The only de…nition of irrational numbers are that they are real, and not rational. Big help.
We know that rational numbers have this de…nition:
They can written as:
p
Q =
: p; q 2 Z; q 6 = 0; and p=q must be in lowest terms
q
6
We say that 6 =
and that is considered to be in lowest terms. [Integers have ‘ 1’as their
1
denominators.]
Thus, 4=8 is a rational number because its lowest terms form is 1/2.
If we were to list the rational numberas in roster form, we, of course, would never list
4=8; because 1/2 would already be in the list somewhere else.
(#2) Concept
p
Let P = “x =
2” .
Let Q = “x is irrational” .
We want to prove:
P ) Q:
Contrapositive FAILS.
(
Q) ) (
P )
p
“Prove that if x is rational, then x 6 =
2:” Again, this is not helpful.
p
We would need to …nd a rule for all rational numbers and then show that
2 is not a member.
Are we stuck? NO.
(#3) Given that P is true, then either Q is true or (
Q) is true.
[Remember that it’ s possible that P may, in fact, have nothing to do with Q: In that case,
P will not imply either one. Theorems are only valid when P is inside Q in the Venn diagram.]
p
We know that
2 is either rational or irrational, but it cannot be both.
Assume that the theorem is FALSE. What is equivalent to
(P ) Q)?
Again, we need that disjunction de…nition!
(
P _ Q) , P ^ (
Q)
So if we assume that the theorem is false, then we assume that P is true and the conclusion
Q is false (even though Q is really true).
Eventually, we must arrive at some contridiction, because:
(
Q) ^ Q = F alse!
1
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education