# Need math help with the step by step solutions for the problems below

**Question description**

Hi Borys:

Here are the problems that I need solutions for:

1) How is this solved? Identify 4 Distinct infinite sets of real numbers A, B, C, and D so that each set is contained in the next one and so that A matches B, B matches C, but does not match D. Use standard sets of numbers such as NAT or RAT etc. as choices for A, B, C, and D. Justify each conclusion.

2) Suppose that a right triangle has sides a, b, and c, which happens to be a natural (whole) numbers, with c as the largest. If only a is an even number, can it be determined whether the perimeter (a + b + c) is odd or even? How is the conclusion justified? How would the answer change if (if any) if a had been an odd nymber?

3) Show how to construct a Golden triangle whose width is 4. What is the length of this of this rectangle? What is the diagonal measurement of this rectangle? Call L/W ( length divided by width) for this rectangle G and call W/L for this rectangle g. Show that the numbers G and g satisfy G = g + 1.

4) Let D denote the set of all real numbers that can be written as a decimal point followed by some combination of the digits 3, 5, and 7. For example: .33333…, .., 77777…, 55555…, and 375375375…are among the numbers in set D. Notice that set D is contained in but not = to the interval [0,1]. Do you believe it is possible to match the set D with the set of natural numbers, i.e., NAT = {1,2,3…}? Justify the answer.

5) Divide the binary number 10,000 by the binary number 111 showing the division. Show a check for the work in binary and by converting to the decimal system.

6) Explain the “Continuum Hypothesis.”

7) Convert these repeating decimals to fractions; (a) 7.575757... (b) .959595…; and, (c) .1111111. Show how this is done.

8) Consider the capital letters of the alphabet (in
simplest form). Then determine the first three such letters which can__ not__
be “traced” (as networks). Next determine the last three which can __not__
be “traced” (as networks). Explain how the theory (Euler’s) justifies the
answers.

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