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Formal Methods Key to Homework Assignment 3, Part 2
• Prove that multiplication of rational numbers is
welldefined.
Proof. Suppose m, n, p, q, r, s, t and u are integers such that n, q, s,
and u are nonzero,
m/n = r/s, and p/q = t/u. We want to see that
or, equivalently
From the definition of equality of rational numbers , we need to see that
mpsu = nqrt.
But by assumption m/n = r/s and p/q = t/u. So ms = nr and
pu = qt. Substituting
into the lefthand side of the equation mpsu = nqrt, we get
nrqt = nqrt.
Reversing the steps , then, we see that
and multiplication of rational numbers is well defined.
89. Is the converse of “if n is any prime, then 2^{n} + 1 prime” true? If your
answer is yes,
prove the statement. Otherwise find a counterexample.
The converse is “if 2^{n} + 1 is prime, then n is prime.” This is false. For
example,
2^{4} + 1 = 17, but 4 isn’t prime.
94. Prove that if a and b are rational numbers with a < b, then there exists a
rational
number r such that a < r < b.
Proof. Define r = (a+b)/2. Then if m, n, p, and q are
integers with n and q nonzero,
such that a = m/n and b = p/q, we have
Since mq + np is an integer and 2nq is a nonzero integer,
we see that r is a rational
number.
To check that a < r < b, we can check that the differences r − a and b − r are
both
positive. We have that
Since a < b, we know that b−a > 0. So r −a > 0 and a < r.
The argument that b−r
is positive is entirely analogous:
which, as we’ve just seen is positive. So r is a rational
number such that a < r < b.
95. Prove that if x is a positive real number , then x + 1/x ≥ 2.
Proof. This is similar to problem 80, which we proved by contradiction. So let’s
try
assuming the contrary. That is, we assume that x is a positive real number such
that
x+1/x < 2. Since x is positive, we can multiply both sides of this inequality by
x and
get
x^{2} + 1 < 2x,
or, equivalently ,
x^{2} − 2x + 1 = (x − 1)^{2} < 0.
However, since x is a real number (x − 1)^{2} ≥ 0, and this is
a contradiction. So the
assumption that x + 1/x < 2 must be false, and x + 1/x ≥ 2 for all positive real
numbers x.
96. (a) Find positive real numbers x and y such that
.
If x = y = 1, then x and y are positive, and
but .
Since 2 is rational and is irrational,
.
(b) Prove that if x and y are positive real numbers, then
.
First Proof. We’ve proved a number of inequalities involving real numbers by
using contradiction. So let’s try contradiction. So we assume that x and y are
positive real numbers such that
Since the function f(z) = z^{2} is increasing on the positive reals, we know that if
0 < a < b then 0 < a^{2} < b^{2}. So
Simplifying , gives
and subtracting x + y from both sides gives
but since x and y are positive reals,
. So ,
which is a contradiction.
So the assumption that
must be false, and
we have
that , for all
positive real numbers x and y.
Second Proof. We can reverse the steps in the argument given in the first proof
and give a direct proof of the the result. Suppose that x and y are positive
real
numbers. Then , and
. Thus
or, equivalently,
Since the function is
increasing on the positive reals, we can take
square roots of both sides of this inequality, and get
Note. Since , this proof actually shows that
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