Homework #4
Finite and Infinite Sets
Read section 1.6
Exercises from the text: Pages 42-3: #1,#4, #8
Other exercises in thought and logic that you will most certainly find interesting.
1. Prove that if A is a countably infinite set, then P(A) is uncountable.
2. Prove that if E is a countable set of points in the plane, then E  A B where A
intersects each horizontal line in finitely many points and B intersects each vertical
line in at most finitely many points.
3. Prove that the Cantor Middle-Third Set is uncountable.
For problems 4 and 5, determine whether the statement is true or false. If true, prove the
statement. If false, give a counterexample.
4. There is a set that is uncountable, but is not equivalent to
.
5. Any set of real numbers that is uncountable contains an open interval.
6. A real number is said to be algebraic if it is a root of a polynomial
a0  a1 x  a2 x 2 
 an x n , where an  0, n  1 , and ai is and integer for every i.
The set of algebraic numbers is uncountable.