Math Problem, Mathematics

Instructions:
1) Write up your solutions on separate sheets of paper, and include all relevant work leading up to your solution. You are being graded on your solutions, not just on your answers ; a “5” will not be awarded merely for having the right answer. Your solutions should be written as though they were to appear in a lucidly written text book, with complete sentences, correct grammar, and appropriately displayed equations (not cramped). Use this sheet as a cover sheet for your test.
2) Your work should be your own. You may consult your notes (including past assignments and my solutions thereto), any resources I have posted to Moodle, and you may see me for help, but you are not permitted to work with each other, and you are not permitted to consult any other resources, including the internet. Your signature below indicates that you have adhered to to these rules.
Name Signature
In the following problems, if A is a set, then its complement is denoted by A0. If a is an element of a Boolean algebra, we denote its dual by a0.
1) If n is an integer such that n = 4k + 1 for some k ? Z, prove that n2 – 1 is divisible by 8.
2) Prove that if 3n + 7 is odd, then n is even.
3) Prove that n3 is even if and only if n is even.
4) Prove that v3 2 is irrational. Hint: you will need to use the result from problem 3.
5) Provide a counterexample to show that the following statement in set theory is generally false: “if A n B = A n C , then B = C ”.
6) Let A and B be subsets of a common universal set U . Prove that if A ? B, then B0 ? A0.
7) Consider the Boolean expression abc + ab0c + ab0c0.
a) Draw the circuit for this expression.
b) Simplify the expression and draw the circuit for the simplified expression. Hint: you can write the expression/circuit with just three gates.
8) Find a binary representation for the base 10 number 2014.
9) Find a binary representation for the base 10 number 1 .