Axiom 3.3. (Singleton sets and pair sets). If \(a\) is an object, then there exists a set \(\{a\}\) whose only element is \(a\), i.e., for every object \(y\), we have \(y\in\{a\}\) if and only if \(y=a\); we refer to \(\{a\}\) as the singleton set whose element is \(a\). Furthermore, if \(a\) and \(b\) are objects, then there exists a set \(\{a,b\}\) whose only elements are \(a\) and \(b\); i.e., for every object \(y\), we have \(y\in\{a,b\}\) if and only if \(y=a\) or \(y=b\); we refer to this set as the pair set formed by \(a\) and \(b\).
Axiom 3.3
- Exercise 3.4.8
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Exercise 3.2.2
\(\Vdash\) { axiom 3.3 }
Use the axiom of regularity (and the singleton set axiom) to show that if \(A\) is a set, then \(A \in A\). Furthermore, show that if \(A\) and \(B\) are two sets, then either \(A \notin B\) or \(B \notin A\) (or both).
- Exercise 3.2.1
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Exercise 3.1.2
Exercise 3.1.2 Using only Definition 3.1.4, Axiom 3.1, Axiom 3.2, and Axiom 3.3, prove that the sets \(\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\), and \(\{\emptyset, \{\emptyset\}\}\) are all distinct (i.e, no two of them are equal to each other).
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Chapter3
Exercise 3.1.2 Using only Definition 3.1.4, Axiom 3.1, Axiom 3.2, and Axiom 3.3, prove that the sets \(\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\), and \(\{\emptyset, \{\emptyset\}\}\) are all distinct (i.e, no two of them are equal to each other).