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the inequalities 0 u 1 and 0 v 1. Since u 1 and v 0 we know that uv 1v which tells us that
uv v. Since uv v and v 1 we conclude that uv 1.
Now since 0 v and u 0 we know that 0u uv which tells us that 0 uv. Therefore 0 uv 1.
4. In this exercise, if we are given three nonnegative integers a, b and c then the integer that consists of a
hundreds, b tens and c units will be written as a, b, c . Given nonnegative integers a, b and c, prove the
assertion P Q R S where P, Q, R and S are, respectively, the following assertions
P. If the number a, b, c is divisible by 3 then the number a b c is also divisible by 3.
9 Q. If the number a b c is divisible by 3 then the number a, b, c is also divisible by 3.
R. If the number a, b, c is divisible by 9 then the number a b c is also divisible by 9.
S. If the number a b c is divisible by 9 then the number a, b, c is also divisible by 9. Hint: In this exercise we are actually given three nonnegative integers a, b and c and so there is no
need to quantify them. In order to prove the assertion P Q R S we have to show that all of the
statements P, Q, R, and S are true. We therefore have four separate proofs to write. We show the proof
of statement Q here and leave the rest to you:
Suppose that the number a b c is divisible by 3. We need to show that the number
a, b, c 100a 10b c
b
is also divisible by 3. Now since the number a3c is an integer and since
a, b, c 100a 10b c 99a 9b a b c
3 33a 3b a b c
3
we know that a, b, c has a factor 3. Some Exercises on Statements Containing “Or”
1. Given that m and n are integers and that the number mn is not divisible by 4, prove that either m is odd or n is
odd. Solution: The required statement can be proved by showing that if both m and n are even then the
number mn has a factor 4. Suppose that both m and n are even. We know that the numbers m and n are
2
2
integers and that
n
mn 4 m
2
2
from which it follows that mn has a factor 4.
Instead of making a new problem out of this exercise we can interpret it as asking us to prove that
if both m and n are even then mn has a factor 4. We did this problem in the 3.2.4.
2. Given that m and n are integers, that neither m nor n is divisible by 4 and that at least one of the numbers m
and n is odd, prove that the number mn is not divisible by 4. Solution: We can interpret this exercise as asking us to prove the following three assertions:
a. If m and n are even then mn must have a factor 4.
b. If m and n are integers and m has a factor 4 then mn has a factor 4.
c. If m and n are integers and n has a factor 4 then mn has a factor 4.
The first of these was handled in the previous exercise and also in an earlier exercise. The other two are
easy and, of course, they are analogs of one another so we need not do both of them.
3. Prove that if x cos 40 or x cos 80 then
8x 3 6x 1 0. Solution: We have two jobs to do:
We need to show that if x cos...
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 Fall '08
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 Math, Calculus

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