Practice problems

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.

b. Explain to Jim how he should decide what to put in his 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...
...

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...
...

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.

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)

Answers