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Math 7350 Homework Assignment 1 Fall 2004
Due September 16.
- Let Ai,j be subsets of a set X for i, j โ N. Show that โ^ โ
i=
โ^ โ
j=
Ai,j =
(ai)
โ^ โ
i=
Ai,ai
where the second union is over all sequences (ai)โ i=0 of natural numbers.
- Let C be a collection of subsets of a set X, and let A be the algebra generated by C. Show that the ฯ-algebra generated by C is equal to the ฯ-algebra generated by A.
- Give an example of a partial order with a unique minimal element, but no smallest element.
- Assume A and B are well ordered sets, with well orderings โค and โคโฒ^ respectively. If A is order isomorphic to an initial segment of B and B is order isomorphic to an initial segment of A, show that A and B are order isomorphic. [Hint: show that the composition of the two order isomorphisms is the identity.]
- We call a set X Dedekind finite if every injection f : X โ X is also surjective. (a) Show that if X is not Dedekind finite, then there is an injection g : N โ X. [Hint: If x 0 โ/ f [X], xn+1 = f (xn), and f is injective, then the xn are distinct.] (b) Show that if there exists an injection g : N โ X then X is not Dedekind finite.
Math 7350 Homework 1 Solutions Fall 2004
- Let Ai,j be subsets of a set X for i, j โ N. Show that โ^ โ
i=
โ^ โ
j=
Ai,j =
(ai)
โ^ โ
i=
Ai,ai
where the second union is over all sequences (ai)โ i=0 of natural numbers.
Assume x โ
i=
j=0 Ai,j^. Then, for all^ i,^ x^ โ^
j=0 Ai,j^. Hence, if we fix^ i, there is a j such that x โ Ai,j. Set ai to be the smallest such j. Now x โ Ai,ai for all i, so x โ
i=0 Ai,ai^. In particular,^ x^ โ^
(ai)
i=0 Ai,ai^.
Now assume x โ
(ai)
i=0 Ai,ai.^ Then there exists a sequence (ai)
โ i=0 such that x โ
i=0 Ai,ai , so^ x^ โ^ Ai,ai for all^ i.^ But then, for all^ i,^ x^ โ^
j Ai,j^.^ Hence x โ
i=
j=0 Ai,j^.
By Extensionality, the two sets are equal.
- Let C be a collection of subsets of a set X, and let A be the algebra generated by C. Show that the ฯ-algebra generated by C is equal to the ฯ-algebra generated by A.
Let ฯ(C) be the ฯ-algebra generated by C and let ฯ(A) be the ฯ-algebra generated by A. Since any ฯ-algebra is also an algebra, ฯ(C) is an algebra containing C, and hence contains A, the smallest algebra containing C. Now ฯ(C) is a ฯ-algebra containing A, so contains ฯ(A), the smallest such ฯ-algebra. Similarly ฯ(A) is a ฯ-algebra that contains A, and hence C, and so contains ฯ(C). Thus ฯ(C) = ฯ(A).
- Give an example of a partial order with a unique minimal element, but no smallest element.
Let X = Z โช {?} with the ordering given by the usual ordering on Z,? โค ?, but? unrelated to any element of Z. Since the usual ordering on Z is a partial ordering, we only need to check the axioms involving ?, which are trivial. Now? is clearly minimal since x โค? implies x = ?. On the other hand, no x โ Z is minimal since x โ 1 < x. Finally, X has no smallest element, since the minimal element? is not โค every other element (or indeed any other element).
- Assume A and B are well ordered sets, with well orderings โค and โคโฒ^ respectively. If A is order isomorphic to an initial segment of B and B is order isomorphic to an initial segment of A, show that A and B are order isomorphic. [Hint: show that the composition of the two order isomorphisms is the identity.]
