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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.**More:**

**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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This question was answered on: * Feb 21, 2020 *

* * Solution~0001100160.zip (18.37 KB)

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