Mathematical Proof and the Principles of Mathematics/Sets/Power sets

Power sets
Power sets allow us to discuss the class of all subsets of a given set $$A$$, i.e. $$\{U \;|\; U \subseteq A\}$$. That this is a set is the subject of the Power Set Axiom.

Axiom "Given a set $A$ there exists a set of sets $S$ such that $U \in S$ iff $U \subseteq A$."

Theorem Given a set $$A$$, there exists a unique set whose elements are the subsets of $$A$$.

Proof If $$S_1$$ and $$S_2$$ are two such sets of subsets then $$U \in S_1$$ if and only if $$U \subseteq A$$. But the same is true of $$S_2$$. Thus $$U \in S_1$$ iff $$U \in S_2$$, and so $$S_1 = S_2$$ by the Axiom of Extensionality. $$\square$$

Definition Given a set $$A$$, the set of all subsets of $$A$$ is called the power set of $$A$$. It is denoted $$\mathcal{P}(A)$$.

Example If $$A = \{a, b, c\}$$ then $$\mathcal{P}(A) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\}$$.

Cartesian products
Recall the Kuratowski definition of an ordered pair, $$(a, b) = \{\{a\}, \{a, b\}\}$$ for $$a$$ and $$b$$ elements of a set $$A$$. Note that $$\{a\}$$ and $$\{a, b\}$$ are both subsets of $$A$$, i.e. they are elements of the power set $$\mathcal{P}(A)$$.

This means that $$(a, b)$$ is a subset of $$\mathcal{P}(A)$$, i.e. $$(a, b) \in \mathcal{P}(\mathcal{P}(A))$$.

We can generalise this slightly with a simple trick. We can define $$(a, b)$$ with $$a \in A$$ and $$b \in B$$ for sets $$A$$ and $$B$$. In order to do this, we simply take the elements $$a$$ and $$b$$ from the union of sets $$A\cup B$$.

In other words, we have $$(a, b) \in \mathcal{P}(\mathcal{P}(A\cup B))$$ with $$a \in A$$ and $$b \in B$$.

Theorem The class of all ordered pairs $$(a, b)$$ of elements of $$A\cup B$$ with $$a \in A$$ and $$b \in B$$, is a set.

Proof The set in question is given by $$\{x \in \mathcal{P}(\mathcal{P}(A\cup B)) \;|\; x = (a,b), a \in A \;\mbox{and}\; b \in B\}$$. This is a set by the axioms of Power Set, Union and the Axiom Schema of Comprehension. $$\square$$

Definition The set of ordered pairs $$(a, b)$$ with $$a \in A$$ and $$b \in B$$ is called the cartesian product of $$A$$ and $$B$$, and is denoted $$A\times B$$.

Exercises

 * Show that for sets $$A, B, C, D$$ we have $$(A\cap B)\times (C\cap D) = (A\times C)\cap (B\times D)$$.


 * Show that for sets $$A, B, C$$ we have $$A\times (B\cap C) = (A\times B)\cap (A\times C)$$.


 * Show that for sets $$A, B, C$$ we have $$A\times (B\cup C) = (A\times B)\cup (A\times C)$$.


 * Show that for sets $$A, B, C$$ with $$A \subseteq B$$ we have $$A\times C \subseteq B\times C$$.