# Introductory real analysis: exercise 3:

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 $a, b \in M$, there is an element $c \in M$ such that $a < c, b. 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 $c \in M$ such that $c \leq a, c \leq b$ and there is no element $d \in M$ such that $a < d \leq a, d \leq b$. Similarly, by the least upper bound of a and b, we mean an element $c \in M$ such that $a \leq c, b \leq c$ and there is no element $d \in M$ such that $a \leq d . 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 $M_{1}, M_{2}, M_{3}$ are ordered sets, then $(M_{1}+M_{2})+M_{3}=M_{1}+(M_{2}+M_{3})$, $(M_{1}.M_{2}).M_{3}=M_{1}.(M_{2}.M_{3})$ 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 $\omega + n$, $\omega + \omega$, $\omega + \omega + n$, $\omega + \omega + \omega$, $\ldots$.

Show that the sets are all countable.

Problem 8:

Construct well-ordered sets with ordinals $\omega . n$, $\omega^{2}$, $\omega^{2}.n$, $\omega^{3}$, $\ldots$.

Show that the sets are all countable.

Problem 9:

Show that $\omega + \omega = \omega. 2$, $\omega + \omega + \omega = \omega. 3$, $\ldots$

Problem 10:

Prove that the set $W(\alpha)$ of all ordinals less than a given ordinal $\alpha$ 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 $\aleph_{1}$ be the power of the set M in the previous problem. Prove that there is no power m such that $\aleph_{0} .

More later,

Nalin Pithwa.

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