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Unformatted text preview: BC satisfying the the condition
AB 2 AC 2 BC 2 .
must have a right angle at C.
The contrapositive: Every triangle ABC that fails to satisfy the the condition
AB 2 AC 2 BC 2 .
will not have a right angle at C.
The denial: There is at least one triangle ABC that has a right angle at C and for which the condition
AB 2 AC 2 BC 2 .
does not hold.
d. If some cats scratch, then all dogs bite.
The converse: If all dogs bite then some cats scratch.
The contrapositive: If there exists a dog that does not bite then no cats scratch.
The denial: Some cats scratch but not all dogs bite.
e. It is with regret that I inform you that someone in this room is smoking.
The converse: This statement doesn’t have a converse.
The contrapositive: This statement doesn’t have a contrapositive form.
The denial: It is without regret that I inform you that someone in this room is smoking. 6 f. If a function is differentiable at a given number then it must be continuous at that number.
The converse: If a function is continuous at a given number then it must be differentiable at that
number.
The contrapositive: If a function is not continuous at a given number then it cannot be differentiable at
that number.
The denial: There exists a function and a real number such that the function is differentiable at the
number but fails to be continuous at that number.
g. Every boy or girl alive is either a little liberal or else a conservative.
The converse: Every little liberal and every conservative is a living boy or girl
The contrapositive: Any individual who fails to be either a little liberal or a conservative cannot be
either a living boy or a living girl.
The denial: There is at least one living boy or girl who is neither a little liberal nor a conservative.
2. In each of the following exercises, write down a denial of the given statement.
a. All cats scratch and some dogs bite.
The denial: Either there is a cat that does not scratch or no dogs bite.
b. Either some cats scratch or if all dogs bite then some birds sing.
The denial: No cats scratch and all dogs bite and no birds sing.
c. He walked into my office this morning, told me a pack of lies and punched me on the nose.
The denial: Either he did not walk into my office this morning or he did not tell me a pack of lies or he
did not punch me on the nose.
d. No one has ever seen an Englishman who is not carrying an umbrella.
The denial: At least one person has seen an Englishman who is not carrying an umbrella.
e. For every number x there exists a number y such that y x.
The denial: There exists a number x such that for every number y we have y x. 3. In each of the following exercises we assume that f and g are given functions. Write down a denial of each of
the following statements:
a. Whenever x 50, we have f x g x .
The denial: There exists a number x 50 such that f x gx. b. There exists a number w such that f x g x for all numbers x w.
The denial: For every number w there exists a number x w such th...
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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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