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(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.)

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  1. (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.)
  • (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.)

  •  (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 – B
    • A2 (this is the cross product)
  • A  B
  • A  (B – A)

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?

  •  (10 points)  “If a < 5and b < 8 then a +  b < 13”
  1. Write the converse of this statement.  Is the converse true?  Explain.
    1. Write the contrapositive of this statement.  Is the contrapositive true?  Explain.
  2. (10 points ) Express the following in English and then determine if each statement is true:  Explain fully.
    1. $xÎZ (x2 =  9).
  • “ yÎR (y   1)

                        (c)  “xÎR  $yÎR  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.  $yÎR  “xÎR  x + y = 5.  (Warning, see example 4 pg 59)

 
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