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form says that
Q
R P.
10. Write the assertion P QR S as simply as you can in its contrapositive form. The contrapositive form of the assertion P Q R S says that
R S P Q
which we can write as
R
S P
Q.
The latter statement says that if both of the statements R and S are false then at least one of the
statements P and Q is false. 3 Strategies for Writing Proofs
8 Some Exercises on Statements Containing “And”
1. Prove the following assertions:
a. If a, b, c, x and y are positive numbers and a x b and b y c, then a xy c.
The point of this exercise is to emphasize that you don’t alrady have numbers a, b, c, x and y quantified
for you in the statement of the exercise. Solution: You need to begin: Suppose that a, b, c, x and y are positive numbers and that a x b and b y c. Then you can argue that a xy a x y b y c. b. If two integers m and n are both even then mn has a factor 4. Solution: You need to begin: Suppose that m and n are even integers. Since m is even we know
that m/2 is an integer and since n is even we know that n/2 is an integer. Since
n
mn 4 m
2
2
we conclude that mn has a factor 4.
c. If an integer m is even and an integer n has a factor 3 then mn has a factor 6.
You need to begin: Suppose that m and n are integers and that m is even and that n has a
factor 3. We know that the numbers m/2 and n/3 are both integers. Since
n
mn 6 m
2
3
we deduce that mn has a factor 6.
2. “Ladies and gentlemen of the jury” said the prosecutor, “We shall demonstrate beyond a shadow of doubt that
on the night of June 13, 1997, the accused, Slippery Sam Carlisle, did willfully, unlawfully and maliciously
kill and murder the deceased, Archibald Bott by striking him on the head with a blunt instrument”. Outline a
strategy that the prosecutor might use in order to prove this charge. How many separate assertions must the
prosecutor prove in order to carry out his promise to the jury?
Of course this problem isn’t serious in suggesting that Slippery Sam should be found not guilty if
any one of the actions decribed by the prosecutor turns out to be untrue. But, had the prosecutor
been a mathematician he would have agreed that his obligation, in order to convict Slippery Sam,
is to show that Sam’s action was wilful and also that it was unlawful and also that it was malicious
and also that it involved killing Mr. Bott and also that it involved murdering Mr. Bott and also that
the act was performed by striking Mr. Bott on the head and also that the murder weapon was blunt.
If even one of these conditions is found to be false then a mathematician might be inclined to find
Slippery Sam not guilty.
3. One of the basic laws of arithmetic tells us that if a and b are any two numbers satisfying the condition a b
and if x 0 then ax bx. Show how this law may be used to show that if 0 u 1 and 0 v 1 then
0 uv 1. Solution: We begin the proof by quantifying u and v: Suppose that u and v are numbers satisfyin...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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