5. 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.)
6. 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.)
7. Let A = {1,2,3,5} and B = {2,6}, 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) 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 eight? Explain
(b) How many bit strings are there of length eight 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 < .7 and b < .7 then a + b <1”
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) ∃x∈Z (x2 = 4).
(b) ∀ y∈R (y 5) ≥
(c) ∀x∈R ∃y∈R x + y = 3.
11. 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.