Math 440, Fall 2014, Assignment 1

The beginner ... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites.

- P. Halmos

Carefully define the following terms, then give one example and one non-example of each:

1. Cartesian product (of two sets).
2. Power set (of a set).
3. Equinumerous (sets).
4. Countable set.
5. Uncountable set.
6. Cardinality of the continuum.
7. Partial order.
8. Maximal element (of a partially ordered set).
9. Largest element (of a partially ordered set).
10. Chain (in a partially ordered set).

Carefully state the following theorems (you need not prove them):

1. Cantor-Bernstein Theorem.
2. Cantor's Theorem.
3. Continuum Hypothesis (of course this is not a theorem, though it is sometimes taken as an axiom).
4. Axiom of Choice (see above).
5. Zorn's Lemma.

Solve the following problems:

1. Prove Cantor's Theorem (exercise 1I.1 contains many hints).
2. Problems 1E and 1H (you will use the results of 1H incessantly for the rest of the semester).

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Cartesian Product of Two Sets:

   Let \(A\) and \(B\) be sets. The cartesian product of \(A\) and \(B\), \(A \times B\), is the set:$$ A \times B = \{(a,b)| a\ in A , b \in B\} $$

   Example:

   Non-Example:

1. Power Set of a Set:

   Example:

   Non-Example:

1. Equinumerous Sets:

   Example:

   Non-Example:

1. Countable Set:

   Example:

   Non-Example:

1. Uncountable Set:

   Example:

   Non-Example:

1. Cardinality of the Continuum:

   Example:

   Non-Example:

1. Partial Order:

   Example:

   Non-Example:

1. Maximal Element of a Poset:

   Example:

   Non-Example:

1. Largest Element of a Poset:

   Example:

   Non-Example:

1. Chain in a Poset:

   Example:

   Non-Example: