Confusion in the equivalence relation statement

Question

In my classical analysis book, Chapter 1, it is written that:

Equivalence relation $(=)$ must be:

$E_1$ (reflexive) $a=a$

$E_2$ (symmetric) $a=b\ \Rightarrow b=a$

$E_3$ (transitive) $a=b, b=c \Rightarrow a=c$

QUESTION:

(i) It seems to me $E_1, E_2$ and $E_3$ are $(1)$ self evident axioms $(2)$ always true.

Are there any situations when any one or two out of $E_1,E_2,E_3$ being false?

(ii) If no, then should not the must be be replaced with are always?

Answer 1

This is a common source of confusion for students. This textbook is not making statements about the usual equality operator (=) that you normally see in equations such as $1 + 1 = 2$. In fact, I think the authors made a poor choice by using the = symbol at all. They probably would have been better off using a tilde or another symbol which does not have a standard meaning.

Instead, this book is setting out requirements that any equivalence relation is required to satisfy. In other words, it's telling you what the phrase "equivalence relation" actually means.

To review definitions which you've probably seen earlier in the book:

• The Cartesian product $X \times Y$ of two sets ($X$ and $Y$) is the set of all ordered pairs $(x, y)$, where $x$ ranges over all elements of $X$ and $y$ ranges over all elements of $Y$.
• A binary relation (or sometimes just a "relation") is a subset of the Cartesian product. In other words, it's a set consisting of ordered pairs $(x, y)$ where $x \in X$ and $y \in Y$.
• To make things easier to notate, relations are often given symbols. If you have some relation $R$, and its symbol is $\sim$, then you might write $x \sim y$ instead of writing $(x, y) \in R$ (but they both mean exactly the same thing). You might also refer to the relation as a whole as $\sim$ instead of $R$.
• It may be the case that $X = Y$. If you want to have an equivalence relation, then this must be the case.
• An equivalence relation is a particular kind of binary relation, which satisfies the requirements you have listed in your question. The actual equality relation obviously satisfies those requirements, but so does the relation "is parallel to" over the set of lines in the plane (when we take the convention that every line is parallel to itself). Therefore, the relation "is parallel to" is also an equivalence relation.