1. If a network can be traced, then it must have exactly 0 or 2 odd vertices. The first two letters of the alphabet that can not be drawn without lifting the pencil are A and E. However “A” does have two odd vertices, yet it still cannot be traced without re-tracing a part of the network. “E” follows the theorem above. The last two letters that cannot be traced are X and Y. “X” only has one even vertex, while “Y” has one odd vertex. These letters do not follow the network-tracing theorem. The letters bolded below cannot be traced without lifting the pencil or by re-tracing. Of the letters that can not be traced, all of then have “dead end” lines that will make tracing without lifting the pencil or re-tracing impossible.
2. Theorem: If A, B, and C are natural numbers and are side lengths of a right triangle, then the sum of A+B+C is going to be even every time.
4. Denote set B as all real numbers written as a decimal followed by some combination of the digits 0 and 1. Denote set A as all natural numbers. Set B can be similarly defined as a set of binary numbers. For every number, there is a binary equivalent. It is possible to match set B with the set of a natural numbers.
5. I created an odd list of numbers. An even number of the of the list were odd, and an odd number of the list were even. An even number of odd numbers will add up to be even. An od number of even numbers will also add up to be even. An even number plus another even number results in an even number. With the supposed long list of numbers with the property of an even number of them are odd and an odd number of them are even, the sum be even.
6. Set A has an infinite set of real numbers. Every element of set A is an element of set B, and every element of set B is an element of set C. Yet, the three sets do not have to match each other. An infinite set A and an infinite set B do not always match up. This follows one of...