Based on previous two blogs. (Reference: Introductory Real Analysis by Kolmogorov and Fomin, Dover Publishers):

**Problem 1:**Exhibit both a partial ordering and a simple ordering of the set of all complex numbers.

**Problem 2:**What is the minimal element of the set of all subsets of a given set X, partially ordered by set inclusion. What is the maximal element?

**Problem 3: **A partially ordered set M is said to be a directed set if, given any two elements , there is an element such that . Are the partially ordered sets in the previous blog(s) Section 3.1 all directed sets?

**Problem 4: **By the *greatest lower bound *of two elements a and b of a partially ordered set M, we mean an element such that and there is no element such that . Similarly, by the *least upper bound *of a and b, we mean an element such that and there is no element such that . By a *lattice *is meant a partially ordered set any two elements of which have both a greatest lower bound and a least upper bound. Prove that the set of all subsets of a given set X, partially ordered by set inclusion, is a lattice. What is the set theoretic meaning of the greatest lower bound and least upper bound of two elements of this set?

**Problem 5: **Prove that an order preserving mapping of one ordered set onto another is automatically an isomorphism.

**Problem 6: **Prove that ordered sums and products of ordered sets are associative, that is, prove that if are ordered sets, then

,

where the operations + and . are the same as defined in previous blog(s).

*Comment: *This allows us to drop the parentheses in writing ordered sums and products.

**Problem 7: **

Construct well-ordered sets with ordinals

, , , , .

Show that the sets are all countable.

**Problem 8:**

Construct well-ordered sets with ordinals

, , , , .

Show that the sets are all countable.

**Problem 9:**

Show that , ,

**Problem 10:**

Prove that the set of all ordinals less than a given ordinal is well-ordered.

**Problem 11:**

Prove that any non-empty set of ordinals is well-ordered.

**Problem 12:**

Prove that the set M of all ordinals corresponding to a countable set is itself uncountable.

**Problem 13:**

Let be the power of the set M in the previous problem. Prove that there is no power m such that .

More later,

Nalin Pithwa.

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