MATH 74 HOMEWORK 4 (DUE WEDNESDAY SEPTEMBER 26)
1.
(c).
[Interesting note:
as of this writing, I do not think it is known whether the
statement is true or false.]
2.
(d).
3.
Suppose the statement is false.
Then there is a rational number
x
and an
irrational number
y
with the property that the number
z
=
x
+
y
is rational. But
then
y
=
z

x
, and a difference of rational numbers is rational. This contradicts
the fact that
y
is irrational, proving the statement.
4.
Suppose the statement is false. Then there is a nonzero rational number
x
and
an irrational number
y
with the property that the number
z
=
xy
is rational. But
then
y
=
z
x
(which makes sense as
x
is nonzero) is a quotient of rational numbers.
We know that a quotient of rational numbers is rational, so
y
is rational.
This
contradicts the fact that
y
is irrational, proving the statement.
5.
If the assertion is false, then there are unequal positive integers
x
and
y
satisfying
x
2
+
xy
= 2
y
2
. But then
x
2
+
xy

2
y
2
= 0, and factoring we conclude
(
x

y
)(
x
+ 2
y
) = 0
.
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 Fall '07
 COURTNEY
 Math, Prime number, Rational number

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