Math 440, Fall 2014, Assignment 1
From cartan.math.umb.edu
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:
- Cartesian product (of two sets).
- Power set (of a set).
- Equinumerous (sets).
- Countable set.
- Uncountable set.
- Cardinality of the continuum.
- Partial order.
- Maximal element (of a partially ordered set).
- Largest element (of a partially ordered set).
- Chain (in a partially ordered set).
Carefully state the following theorems (you need not prove them):
- Cantor-Bernstein Theorem.
- Cantor's Theorem.
- Continuum Hypothesis (of course this is not a theorem, though it is sometimes taken as an axiom).
- Axiom of Choice (see above).
- Zorn's Lemma.
Solve the following problems:
- Prove Cantor's Theorem (exercise 1I.1 contains many hints).
- Problems 1E and 1H (you will use the results of 1H incessantly for the rest of the semester).
Questions:
Solutions:
Definitions:
- 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:
The cartesian product of \(\{a,b\}\), and \(\{1,2\}\) is \(\{(1,a), (1,b), (2,a), (2,b)\}\)
Non-Example:
Remember that the Cartesian product is a set of tuples. Don't confuse it with the union of two sets, which would be \(\{1,2,a,b\}\) for the above example.
- Power Set of a Set:
Example:
Non-Example:
- Equinumerous Sets:
Example:
Non-Example:
- Countable Set:
Example:
Non-Example:
- Uncountable Set:
Example:
Non-Example:
- Cardinality of the Continuum:
Example:
Non-Example:
- Partial Order:
Example:
Non-Example:
- Maximal Element of a Poset:
Example:
Non-Example:
- Largest Element of a Poset:
Example:
Non-Example:
- Chain in a Poset:
Example:
Non-Example:
