This preview shows page 1. Sign up to view the full content.
Unformatted text preview: at f x gx. c. For every number x there exists a number 0 such that for every number t satisfying the condition
x t , we have f x f t  1.
The denial: There exists a number x such that for every number 0 there is at least one number t
satisfying the condition x t  such that f x f t  1.
d. There exists a number 0 such that for every pair of numbers x and t satisfying the condition
x t , we have f x f t  1.
The denial: For every number 0 it is possible to find a pair of numbers x and t satisfying the
condition x t  and for which f x f t  1. 7 e. For every number 0 and for every number x, there exists a number 0 such that for every number
t satisfying x t , we have f t f x  .
The denial: There exists a number 0 and a number x such that for every number 0 it is
possible to find a number t such that x t  and f x f t 
.
f. For every positive number there exists a positive number such that for every pair of numbers x and t
satisfying the condition x t , we have f x f t  .
The denial: There exists a number 0 such that for every number 0 it is possible to find a pair
of numbers x and t satisfying the condition x t  for which f x f t 
.
4. Explain why the statement P Q is equivalent to the statement P Q. The assertion P Q says that if P is true then Q must also be true. This assertion says nothing at
all about Q in the event that P is false. The only way in which the assertion P Q can be false is
that P is true and Q is not. In other words, the denial of the condition P Q says that P
Q.
5. Explain why the statement
is false and Q is true). P Q is equivalent to the assertion that either (P is true and Q is false) or (P The assertion P Q says that P and Q have the same truth value. So the assertion
that they don’t, which means that one of them is true and the other is false.
6. Explain why the statement P Q is equivalent to the statement P P Q says Q. The assertion P Q says that at least one of the statements P and Q is true. So the denial of the the
assertion P Q says that they are both false.
7. Explain why the statement P Q
statements Q and R are false. R is equivalent to the assertion that P is true and that both of the The denial of the assertion P Q R says that P is true but that the assertion Q R is false. In
other words, it says that P is true and that both of the statements Q and R are false.
8. Explain why the converse of the statement P Q R is equivalent to the condition R P QP. The converse of the statement P Q R says that Q R P and this says that P must be true
if at least one of the statments Q and R are true. In other words, the satement Q R P says
that Q P and also that R P.
9. Write the assertion P Q R as simply as you can in its contrapositive form. The contrapositive form of the assertion P Q R says that
Q R P and this says
that if both of the statements Q and R are false then P is false. In other words, this co...
View
Full
Document
 Fall '08
 STAFF
 Math, Calculus

Click to edit the document details