- #1
airbusman
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Here it is, for you to critique. This is a proof by contradiction. This is a good example of how I usually go about doing proofs, so if you give me tips on how to improve this particular proof, I'll be able to improve all my other proofs.
I just learned how to do proof by contradiction yesterday, so forgive any ugliness in my proof.
1) Prove: r√2 is irrational, where r is rational and r ≠ 0
variables involved:
r, rational
m, integer
n, integer
a, integer
b, integer
a', integer
b' integer
x, integer
y, integer
Knowledge of the proof that √2 is irrational was used. That is all.
1) r√2 = (m/n)√2 = a/b (this is assuming that Q in the implication P→Q is negated, which is the basis of the proof by contradiction)
2) (m√2)/n = 2a'/2b' = a'/b'
a'/b' = x/y, where at least one of x or y is odd (this is where the contradiction will lie later on)
3) (m√2)/n = x/y
m√2 = xn/y
√2 = xn/ym
2 = ((x^2)(n^2))/((y^2)(m^2))
2y^2m^2 = x^2n^2
.....after solving for x.....
x^2 is even therefore x is even
4) x = 2k
2y^2m^2/n^2 = 4k^2
.....after solving for y....
y^2 is even so y is even
So there is a contradiction; both x and y are even when at least one of x or y should be odd. Therefore, the statement that was required to be proved is true.
I just learned how to do proof by contradiction yesterday, so forgive any ugliness in my proof.
Homework Statement
1) Prove: r√2 is irrational, where r is rational and r ≠ 0
variables involved:
r, rational
m, integer
n, integer
a, integer
b, integer
a', integer
b' integer
x, integer
y, integer
Homework Equations
Knowledge of the proof that √2 is irrational was used. That is all.
The Attempt at a Solution
1) r√2 = (m/n)√2 = a/b (this is assuming that Q in the implication P→Q is negated, which is the basis of the proof by contradiction)
2) (m√2)/n = 2a'/2b' = a'/b'
a'/b' = x/y, where at least one of x or y is odd (this is where the contradiction will lie later on)
3) (m√2)/n = x/y
m√2 = xn/y
√2 = xn/ym
2 = ((x^2)(n^2))/((y^2)(m^2))
2y^2m^2 = x^2n^2
.....after solving for x.....
x^2 is even therefore x is even
4) x = 2k
2y^2m^2/n^2 = 4k^2
.....after solving for y....
y^2 is even so y is even
So there is a contradiction; both x and y are even when at least one of x or y should be odd. Therefore, the statement that was required to be proved is true.