Mathematics 5
Winter Term 2000
The World According to Mathematics
Mid-Term Problem Solving Exam

January 28, 2000

Name:______________________________

In class on Monday, February 7, hand in this sheet with your solutions to the following
problems. You should feel free to ask questions of your instructors as you work on these
problems. However, it is a violation of the honor principle to consult anyone else. You
may use as resources only the text and notes of the course. Good luck.

(4 points)

1.

Let's define an infinite setas a set that can be placed in a one-to-one
correspondence with a proper subset of itself. [Note: Set Ais a proper
subset of set Bif Ais contained in Band not equal to B.] Using this
definition of infinite set, show that {5,10,15, ...} is an infinite set.

(5 points)

2.

Here is an algorithm for testing for divisibility by 11: Find the sum of
the digits in the odd positions (from the right) and the sum of the digits
in the even positions. Then find the difference between the sums. If the
difference is 0 or a number divisible by 11, then the original number is
divisible by 11. For example, to determine whether 9867 is divisible by
11, first form the sum 7 + 8 = 15, then the sum 6 + 9 = 15, and subtract
15 - 15 = 0. Since the difference is 0, the original number 9867 is
divisible by 11. (Of course, we can check this answer by actually
dividing 11 into 9867 to obtain 897.)

(a)
(b)

Use the algorithm to check that 6116 is divisible by 11.
A number that reads the same forwards and backwards is called a
palindrome. The number 6116 in part (a) is a palindrome. Use the
algorithm to explain why every four-digit palindromic number
must be divisible by 11.
Is every five-digit palindromic number divisible by 11? Give an
example to support your answer.
Is every six-digit palindromic number divisible by 11? Explain.
Based on what you have learned in parts (a) through (d), what do
you believe is true about palindromic numbers and why?

(c)

(d)
(e)

(4 points)

3.

Determine whether the following statements are logically equivalent:

(4 points)

4.

Using the fact that p -> q is equivalent to ~p V q, rewrite the following
statements in the alternative form:

(a)

The toothpaste does not contain fluoride or it does not contain
sodium.
If the grass is not green, then it did not rain.

(b)