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120 The Nuts and Bolts of Proofs, Third Edition
THEOREM 8.
The number V2 is irrational.
Alleged Proof. Let us assume that V2 is a rational number. Then
y/l = a/b, with a and b positive integers. Thus,
fc2
Therefore,
a' = 2b\
This impHes that a^ is an even number; therefore, a is an even number.
So, a = lau with a\ integer positive number and a\<a.
This yields Aa\ = 2b^; that is, laj = b^. Therefore, b^ is an even number,
which imphes that b is an even number. So, b =^2bu with bi integer positive
number and bi<b. Thus,
/^ _ fl _ 2^1 _ ai
~b~2bi~bi'
Because \/2 = ai/bi, we can repeat the process above and write ai — 2a2
and bi = Ibi where ai and ^2 are positive integers, b2<b\, and a2<a\.
If this process is repeated k times, we can construct two sequences of
integer positive numbers:
0 < ak < ak-i < '" < ai < a\ < a
0 < bk < bk-i < "' <b2 <b\ <b.
If k> b, we have reached a contradiction. Therefore, v^ is an irrational
number. •
THEOREM 9. Let A, B, and C be any three subsets of a universal set U. Then
AU(BnC) = {AUB)U{An C).
Alleged Proof xeAU(BnC)\f and only if either xeAorxeBDC
if and only if either x £ A or x e B and x^ C if and only if either x G ^ or
X e B or X e A and x^ C if and only if either x e AU B or x e A D C if and
only ifxe{AUB)U(An C). •