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[solution] » Problem 4.7 Determine whether the proposed negation is correct.

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Problem 4.7 Determine whether the proposed negation is correct.
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Problem 4.7

Determine whether the proposed negation is correct. If it is not, write a correct negation.

Proposition : For all integers n, if n2is even then n is even.

Proposed negation : For all integers n, if n2is even then n is not even.


Problem 4.8

Let D = {?48, ?14, ?8, 0, 1, 3, 16, 23, 26, 32, 36}. Determine which of the following propositions are true and which are false. Provide counterexamplesfor those propositions that are false.

a. ?x ? D, if x is odd then x > 0.

b. ?x ? D, if x is less than 0 then x is even.

c. ?x ? D, if x is even then x ? 0.

d. ?x ? D, if the ones digit of x is 2, then the tens digit is 3 or 4.

e. ?x ? D, if the ones digit of x is 6, then the tens digit is 1 or 2


Problem 4.9

Write the negation of the proposition :?x ? IR, if x(x + 1) > 0 then x > 0 orx x.? For each x given below, find an n to make the predicate n > xtrue.

a. x = 15.83

b. x = 108

c. x = 101010.40 FUNDAMENTALS OF MATHEMATICAL LOGIC


Problem 4.12

Given the proposition: ?x ? IR, ? a real number y such that x + y = 0.

a. Rewrite this proposition in English without the use of the quantifiers.

b. Find the negation of the given proposition.


Problem 4.13

Given the proposition: ?x ? IR, ?y ? IR, x + y = 0.

a. Rewrite this proposition in English without the use of the quantifiers.

b. Find the negation of the given proposition.


Problem 4.14

Consider the proposition ?Somebody is older than everybody.? Rewrite thisproposition in the form ?? a person x such that ? .?


Problem 4.15

Given the proposition: ?There exists a program that gives the correct answerto every question that is posed to it.?

a. Rewrite this proposition using quantifiers and variables.

b. Find a negation for the given proposition.


Problem 4.16

Given the proposition: ?x ? IR, ?y ? IR such that x 0 then x > 0 or x ?1.?


Problem 4.18

Rewrite the following proposition in if-then form : ?Earning a grade of C?in this course is a sufficient condition for it to count toward graduation.?


Problem 4.19

Rewrite the following proposition in if-then form : ?Being on time each dayis a necessary condition for keeping this job.?

 







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