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# The homework I need help with is discrete Structures I. Please see attached.

1. (10 pts.) Prove the following: If n is an odd integer then n 2 is an odd integer. (This is #4, pg. 8, week 3 of the notes.) 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) A 2 (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” a. Write the converse of this statement. Is the converse true? Explain. b. 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) \$ Z (x 2 = 9). (b) " R (y 1) (c) " R \$ R x + y = 3. 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. \$ R " R x + y = 5. (Warning, see example 4 pg 59)
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