91Ó°ÊÓ

In this exercise you are asked to write up some of the axioms of Zermelo- Fraenkel set theory, also known as ZF. The language of set theory consists of a single binary relation symbol, \(\in\), that is intended to represent the relation "is an element of". So the formula \(x \in y\) will usually be interpreted as meaning that the set \(x\) is an element of the set \(y\). Here are English versions of some of the axioms of ZF. Write them up formally as sentences in the language of set theory. The Axiom of Extensionality: Two sets are equal if and only if they have the same elements. The Null Set Axiom: There is a set with no elements. The Pair Set Axiom: If \(a\) and \(b\) are sets, then there is a set whose only elements are \(a\) and \(b\). The Axiom of Union: If \(a\) is a set, then there is a set consisting of exactly the elements of the elements of \(a\). [Query: Can you figure out why this is called the axiom of union? Write up an example, where \(a\) is a set of three sets and each of those three sets has two elements. What does the set whose existence is guaranteed by this axiom look like?] The Power Set Axiom: If \(a\) is a set, then there is a set consisting of all of the subsets of \(a\). [Suggestion: For this axiom it might be nice to define \(\subseteq\) by saying that \(x \subseteq y\) is shorthand for (some nice formula with \(x\) and \(y\) free in the language of set theory).]