Practice problems solutions

1. Mike and Jim are playing the X-O game with rows that are 3-long.  Mike goes first;

Mike's board:








Jim's board


a. Explain to Jim how he should decide what to put in his first square.
    Whatever Mike puts in the first square of the first row, put the opposite thing in your first square.

b. Explain to Jim how he should decide what to put in his second square.
    Whatever Mike puts in the second square of the second row, put the opposite thing in your second square.

2. Here is the beginning of an infinite list of decimal numbers between 0 and 1. Show the first 6 digits of a decimal number that would not be on the list using only the digits 1 and 4, and write down the rule that tells how I should continue your pattern to get a decimal that would not be equal to any of the numbers on the (infinite) list:

.1 2 3 4 5 6 7 8...
.1 3 2 2 2 2 2 3...
.4 1 4 1 1 4 1 1...
.5 8 7 9 2 1 3 4...
.7 9 6 2 1 4 5 8...
.5 0 0 0 0 0 0 0...
...

Answer A: .411141...
Look at the n-th digit after the decimal of the n-th number.  If that digit is 1, your n-th digit should be 4.  If that digit is not 1, your n-th digit should be 1

Answer B: .441444...
Look at the n-th digit after the decimal of the n-th number.  If that digit is 4, your n-th digit should be 1.  If that digit is not 4, your digit should be 4.

3. Here is the beginning of an infinite list of decimal numbers between 0 and 10. Write a rule using the digits 4 and 6 that would give a number that was not on the list, and show how to use it to get the first 6 digits of that number

6. 2 9 6 8 6 1...
2. 8 1 9 6 3 5...
1. 4 1 4 2 1 3...
0. 5 3 4 9 2 4...
8 .6 4 6 6 4 6...
2. 6 6 6 6 6 6...
...

Answer A: 4.44644...
Look at the n-th digit of the n-th number (start counting with the digit before the decimal).  If that digit is not 4, use 4 as your n-th digit.  If that digit is 4, use 6 as your digit.

Answer B: 4.66644...
Look at the n-th digit of the n-th number (start counting with the digit before the decimal).  If that digit is not 6, use 6 as your n-th digit.  If that digit is 6, use 4 as your digit.

Answer C: 0.446464...
Look at the n-th digit following the decimal place of the n-th number.  If that digit is not 4, use 4 as your n-th digit.  If that digit is 4, use 6 as your digit.

Answer D: 0.666664...
Look at the n-th digit following the decimal place of the n-th number.    If that digit is not 6, use 6 as your n-th digit.  If that digit is 6, use 4 as your digit.

4. Tell the definition of cardinality: what it means for two infinite sets to be the same size and to not be the same size.

Two infinite sets are the same size if there is a 1-1 correspondence.  That means you can match the elements of the sets up in a way that matches each thing in one set to one thing in the other set and vice versa.

Two infinite sets are different sizes if it's impossible to get a 1-1 correspondence between them.  So, no matter what way you try of matching the elements up, there will always be elements of one set that dont have a matching element in the other set.

5. Explain how your answer to #2 proves that there are more real numbers between 0 and 1 than there are natural numbers (it has a bigger cardinality)


My rule tells how you can always find a real number that didn't get counted/listed, not matter what counting order someone comes up with.