1. Consider the statement, “For all natural numbers n, if n is prime, then n is solitary." You do notneed to know what solitary means for this problem, just that it is a property that some numbershave and others do not.(a) Write the converse and the contrapositive of the statement, saying which is which. Note:the original statement claims that an implication is true for all n, and it is that implicationthat we are taking the converse and contrapositive of.(b) Write the negation of the original statement. What would you need to show to provethat the statement is false?(c) Even though you don't know whether 10 is solitary (in fact, nobody knows this), is thestatement "if 10 is prime, then 10 is solitary" true or false? Explain.(d) It turns out that 8 is solitary. Does this tell you anything about the truth or falsity of theoriginal statement, its converse or its contrapositive? Explain.(e) Assuming that the original statement is true, what can you say about the relationshipbetween the set P of prime numbers and the set S of solitary numbers. Explain.

The given statement is ''For all natural numbers n, if n is prime ,then n is solitary''.

Answered: 1. Consider the statement, “For all…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. For the following statements, write in symbolic language with quantifiers, negate the statement in symbolic language, determine if the statement is true or false and justify your answer with a proof or counterexample. (a) There is a natural number greater than two that is not the sum of two primes. (b) There is a prime that is one more than a multiple of 4 and is also a sum of two squares. (c) For any natural number n, there is a natural numbers that is the product of n with itself. (d) There is a natural number s such that for any natural number n, s is the product of n with itself. (e) If the product of a rational number x and a non-zero real number y is rational, then x is zero or y is rational. (f) If a and b are integers then there are integers m and n such that a = m+n and b=m-n.
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In logic what is the difference between propositional forms that are necessarily equivalent and propositional forms which are equivalent? In chapter 1.1 of book “A Transition to Advanced Mathematics” by Douglas Smith, he states two propositional forms are equivalent if they have the same truth table. The book doesn’t mention anything about “necessary equivalence” until chapter 1.1, exercise 9: “Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and S is equivalent to R. For each pair of forms, determine whether they are necessarily equivalent. If they are equivalent explain why.”
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The following problem is that of Discrete Mathematics. Consider the following implication, where n stands for an unknown integer.“If n is divisible by 10, then n is divisible by 5.” a. Write the equivalent “or” form of this implication as an English sentence. b. Write the logical negation of this implication as an English sentence. (Do not use “It isnot the case that…”)
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