2 T -periodic solution for m order neutral type differential by Zhang B.

By Zhang B.

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Differential Equations & Control Theory

In keeping with papers on the Intl Workshop on Differential Equations and optimum regulate held lately at Ohio collage, Athens.

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8. 9. 10. 3. Suppose that A ⊂ B ⊂ C. What is A\B? What is A\C? What is A ∪ B? Describe the set Q\Z in words. Describe R\Q. Describe Q × R in words. Describe Q × Z. Describe (Q × R)\(Z × Q) in words. Give an explicit description of the power set of S = {a, b, 1, 2}. Let S = {a, b, c, d}, T = {1, 2, 3}, and U = {b, 2}. Which of the following statements is true? a. {a} ∈ S b. 1 ∈ T c. {b, 2} ∈ U 11. Write out the power set of each set: a. {1, ∅, {a, b}} b. { • , , ∂} 12. Prove using induction on k that if the set S has k elements and the set T has l elements then the set S × T has k · l elements.

But then n 2 = n · n = (2m) · (2m) = 4m 2 = 2(2m 2 ) Our calculation shows that n 2 is twice the natural number 2m 2 . So n 2 is also even. We have shown that the hypothesis that n is twice a natural number entails the conclusion that n 2 is twice a natural number. In other words, if n is even then n 2 is even. That is the end of our proof. 1 What is the role of truth tables at this point? Why did we not use a truth table to verify our proposition? One could think of the statement that we are proving as the conjunction of infinitely many specific statements about concrete instances of the variable n; and then we could verify each one of those statements.

We are delivering 11 letters (that is, the randomly selected points) to these 10 mailboxes. By the pigeonhole principle, some mailbox must receive two letters. 1, contains two of the randomly selected points. 1 inch. 4 Proof by Induction The logical validity of the method of proof by induction is intimately bound up with the construction of the natural numbers, with ordinal arithmetic, and with the so-called well-ordering principle. We shall not treat those logical niceties here, but shall instead concentrate on the technique.

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