(10 pts.) Prove the following: If n is an odd integer then n2 is an odd integer. (This is #4, pg. 8, week 3 of the notes.) 1. 2. (10 pts.) Assume that n is a positive integer. Use the proof by contradiction method to prove and explain: If 3n + 2 is an even integer then n is an even integer. (This is #3, pg. 11, week 3 of the notes.) 3. (15 pts.) Let A = {1,3,5,7} and B = {2,7}, with the universal set U = {1,2,3,4,5,6,7,8,9}. Compute (a) A - B (b) A2 (this is the cross product) (c) A B (d) A (B - A) (e) 8. (10 pts.) A bit string is a string of bits (0's and 1's). The length of a bit string is the number of bits in the string. An example, of a bit string of length four is 0010. An example, of a bit string of length five is 11010. Use the Rule of Products to determine the following: (a) How many bit strings are there of length six? Explain (b) How many bit strings are there of length six which begin with a 1 and end with a 0? Explain (c) How many bit strings are there of length eight with even parity (an even number of 1's)? Explain For your information question 8 part a could have been stated the following way. Computers use bit strings of length 8, called bytes, to represent the characters (letters both upper case and lower case, punctuation symbols, [, {, the integers 0 through 9 etc) on a key board. The Extended ASCII code is one such coding system. Some examples of this code are: \"a\" is represented by 01100001, \"A\" is represented by 01000001 and \"{\" is represented by 01111011 and the number 1 is represented by 00000001. How may such symbols can be described using a byte? 9. (10 points) \"If a < 5and b < 8 then a + b < 13\" Write the converse of this statement. Is the converse true? Explain. Write the contrapositive of this statement. Is the contrapositive true? Explain. 10. (10 points ) Express the following in English and then determine if each statement is true: Explain fully. a. b. $xZ (x2 = 9). (b) " yR (y 1) (c) "xR $yR x + y = 3. (a) Extra Credit Problems. Do all, none or any one of the following extra credit problems: 1. Text page 16 number 46. Explain 2. How many bit strings of length n are palindromes? Hint: Consider two cases n is even and n is odd. Note a palindrome is a \"string\" of letters or numbers which read the same \"frontwards\" and backwards\". Examples: MOM, 1101011, 10111101 are palindromes. 3. Express the following in English and determine whether it is true. Explain. $yR "xR x + y = 5. (Warning, see example 4 pg 59)