Test 4 review (updated now with answers/links)

Test 4 is scheduled for Thursday 8 December.

The review problems are here.

Answers /links are below the fold...

1a) Countable sets:  (remember that a countable set must be inifinite!
 The even natural numbers
 
The rational numbers

 The prime numbers

1b) Infinite sets:
 The even natural numbers

 The real numbers

 The rational numbers

 The irrational numbers

 The prime numbers

1c) Finite sets:
 The solutions to the equation x² − 3x + 2 = 0
 The odd natural numbers less than 500


Note: which of the sets in the list are uncountable? (There are only two.)

2) Here is a way to do it:
    Natural numbers:         1  2  3  4  5  6   7   8   9  ...
                                          |   |   |   |   |   |     |    |    |
    Odd natural numbers:  1  3  5  7  9  11 12 13 14 ...
Here is a way to describe the rule: each natural number n corresponds to the odd natural number you get by doubling n and then subtracting 1: n corresponds to 2n-1. (Check that this rule works. There are other possible rules, but whatever you use you should check that it works: that it gives the right results for the numbers you have listed so far.)

3) A number that is not on the list starts out 0.22224...
    We start with the first decimal digit of the first number. If it is 2, change it to 4; if it is not 2, change it to 2.
    Then use this rule with the 2nd decimal digit of the 2nd number, the 3rd decimal digit of the 3rd number, and so on for all the (infinitely many) numbers in the list.


We know that this number is not any of the numbers already on the list, because it differs from the first number in the first decimal place; it differs from the 2nd number in the 2nd decimal place, etc... and in general, it differs from the nth number in its nth decimal place, for any natural number n. So this is a new real number not on the list.

[Note: you do not have to use the exact rule we used in class: but whatever your rule is, must avoid changing digits to either 0 or 9. and other choice of digits to change to will work, but 0 and 9 have a potential problem because of the non-uniqueness of some decimal expansions. You can see an explanation of Cantor diagonalization which uses a different rule in this video.]

4a)  The subsets are {} (the empty set), {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, and {a,b,c}=S


4b) There are 2⁹ = 512 elements in P(T), and there are 2²⁹ = 2⁵¹² elements in P(P(T)).

For Power sets, see Math is Fun

5a) In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides.

There are other, equivalent ways to state it: the crucial thing is that it refers to a right triangle (a triangle with a right angle), and that the hypotenuse is singled out.

[So.... If you just write a² + b² = c² without further comment or explaining what a, b, and c stand for, that is NOT a correct statement of the theorem and you will NOT get any credit for it!]

5b) See this proof from Math is Fun, or any other proof that you can clearly explain.