# the "is an element of" relation and axiom of substitution

In the 2nd edition of the textbook Analysis I by Terence Tao, the equality of sets at the beginning of the set theory chapter was a definition, as followed.

Definition 3.1.4 (Equality of sets). Two sets $$A$$ and $$B$$ are equal, $$A = B$$, iff every element of $$A$$ is an element of $$B$$ and vice versa. To put it another way, $$A = B$$ if and only if every element $$x$$ of $$A$$ belongs also to $$B$$, and every element $$y$$ of $$B$$ belongs also to $$A$$.

One can easily verify that this notion of equality is reflexive, symmetric, and transitive (Exercise 3.1.1). Observe that if $$x \in A$$ and $$A = B$$, then $$x \in B$$, by Definition 3.1.4. Thus the "is an element of" relation $$\in$$ obeys the axiom of substitution (see Section A.7). Because of this, any new operation we define on sets will also obey the axiom of substitution, as long as we can define that operation purely in terms of the relation $$\in$$.

In the third edition and following the changes mentioned here, this particular change was made:

Page 35: Definition 3.1.4 has to be given the status of an axiom (the axiom of extensionality) rather than a definition, changing all references to this definition accordingly. This requires some changes to the text discussing this definition. Firstly, in the preceding paragraph, “define the notion of equality” will now be “seek to capture the notion of equality”, and “formalize this as a definition” should be “formalize this as an axiom”. For the paragraph after Example 3.1.5, delete the first two sentences, and remove the word “Thus” from the third sentence. Exercise 3.1.1 is now trivial and can be deleted.

Now in the 4th edition of the book, the definition is changed to be an axiom:

Axiom 3.2 (Equality of sets). Two sets $$A$$ and $$B$$ are equal, $$A = B$$, iff every element of $$A$$ is an element of $$B$$ and vice versa. To put it another way, $$A = B$$ if and only if every element $$x$$ of $$A$$ belongs also to $$B$$, and every element $$y$$ of $$B$$ belongs also to $$A$$.

and the paragraph is now this:

The "is an element of" relation $$\in$$ obeys the axiom of substitution (see Section A.7). Because of this, any new operation we define on sets will also obey the axiom of substitution, as long as we can define that operation purely in terms of the relation $$\in$$.

Here is the substitution axiom from A.7:

(Substitution axiom). Given any two objects $$x$$ and $$y$$ of the same type, if $$x = y$$, then $$f(x) = f(y)$$ for all functions or operations $$f$$. Similarly, for any property $$P(x)$$ depending on $$x$$, if $$x = y$$, then $$P(x)$$ and $$P(y)$$ are equivalent statements.

So the question is, why was the reasoning behind this removed, and was it because of the transition from definition to axiom? How is the "Exercise 3.1.1 now trivial and can be deleted."? (Exercise 3.1.1 asked to prove reflexivity, symmetry and transitivity of the removed definition of equality)

And why is this paragraph even in the book and mentions the axiom of substitution, while the axiom is clear and concise about what equality is?

The issue, as mentioned, is the substitution property of equality in first-order logic. Formally, for any predicate $$\varphi(x)$$ (i.e., formula) in the first-order language under discussion (here ZF), we have $$x=y\to (\varphi(x)\leftrightarrow\varphi(y))$$ The formula $$\varphi(x)$$ is allowed to have so-called parameters, i.e., other variable in it. For example: $$x=y\to (x\in A\leftrightarrow y\in A)$$ with $$\varphi(x)\equiv x\in A$$, or $$x=y\to (z\in x\leftrightarrow z\in y)$$ with $$\varphi(x)\equiv z\in x$$. You will recognize this second example as the converse of the Axiom of Extensionality: $$(z\in x\leftrightarrow z\in y)\to x=y$$ So if you define the equality relation by the formula $$z\in x\leftrightarrow z\in y$$, you get this second example for free. But then you'll still need an axiom to insure that the first example holds, i.e., that equal sets are elements of exactly the same sets.