Axiom 3.1. (Sets are objects). IF \(A\) is a set, then \(A\) is also an object. In particular, given two sets \(A\) and \(B\), it is meaningful to ask whether \(A\) is also an element of \(B\).
Axiom 3.1
- Exercise 3.4.8
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Exercise 3.3.1
\(\Rightarrow\) { \(X\), \(X'\) and \(X''\) are objects by Axiom 3.1,
\(~~~\) transitive axiom for any three objects of the same type }\(\Rightarrow\) { \(X\) and \(X'\) are objects by Axiom 3.1,
\(~~~\) symmetry axiom for any two objects of the same type }\(\Rightarrow\) { \(Y\), \(Y'\) and \(Y''\) are objects by Axiom 3.1,
\(~~~\) transitive axiom for any three objects of the same type } -
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).