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.