Proof By Contradiction Worksheet Page 3
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1
2
3(#7) We know that if x; y 2 Q; then (x
y) ; xy; and x=y are all rational also.
Easy proofs?
Suppose [blah blah 2 Z]
p
p
1
2
x =
and y =
:
q
q
1
2
The sum is
p
p
p
q
+ p
q
1
2
1
2
2
1
x + y =
+
=
:
q
q
q
q
1
2
1
2
This last expression does NOT need to be in lowest terms, but the point is that it is in the
form of integer/integer, so it can be reduced if necessary, and thus, (x + y) must be rational.
p
(#8) Show
2 + 1 is irrational.
p
BWOC, assume that x =
2 + 1 is rational. Subtract one from both sides.
p
x
1 =
2
The left side is rational since the di¤erence of two rational numbers is rational.
p
()() We know that
2 is irrational.
p
Thus,
2 + 1 is irrational, not rational.
3
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