Raising the bar

Equivalence relation explained

Many students struggle understanding an equivalence relation. The problem with it is that it is a mental concept that heavily relies on the axiomatic approach. It is the teacher's task to give a more visual and tangible (closely related to real-life experience) definition. The definition should be preceded or accompanied by an explanation that awakes the students' intuition.

For any person $latex p,$ denote $latex Fr\left( p\right) $ the circle (set) of that person's friends. We assume that for each $latex p,$ the circle of friends $latex Fr\left( p\right) $ is unique (in reality, depending on the situation, the circle of friends may change, inluding childhood friends, colleagues, neighbors etc.).

Definition 1. The mapping $latex p\rightarrow Fr\left( p\right) $ satisfies the following conditions.

a) We assume, by definition, that any person is a friend of him/herself. Then, obviously, $latex p\in Fr\left( p\right) .$

b) When $latex p$ considers $latex q$ a friend, $latex q$ does not necessarily consider $latex p$ a friend. We exclude such a situation and consider friendship to be a mutual feeling. Then $latex q\in Fr\left( p\right) $ implies $latex p\in Fr\left( q\right) .$

c) We assume that the adage "a friend of my friend is my friend" is true (in reality, my friends often have their own friends whom I do not consider my friends). In terms of our friendship sets this means that $latex q\in Fr\left(p\right) $ and $latex r\in Fr\left( q\right) $ implies $latex r\in Fr\left( p\right) .$

Remark. In fact, it is sufficient to assume that for each $latex p,$ the circle of friends $latex Fr\left( p\right) $ exists, and then by c) we will have its uniqueness. The mapping $latex p\rightarrow Fr\left( p\right) $ is not a function, because $latex Fr\left( p\right) $ can consist of more than one person. Such a mapping is called a correspondence.

Statement. 1) The circles of friends $latex Fr\left( p\right) $ cover all people. 2) Any two circles $latex Fr\left( p\right) ,$ $latex Fr\left( q\right) $ either coincide, $latex Fr\left( p\right) =$ $latex Fr\left( q\right) $ or do not intersect, $latex Fr\left( p\right) \cap Fr\left( q\right) =\varnothing $ (the situations $latex Fr\left( p\right) \backslash Fr\left( q\right) \neq \varnothing $ and $latex Fr\left( q\right) \backslash Fr\left( q\right) \neq \varnothing $ are impossible).

Proof. 1) is obvious because $latex p\in Fr\left( p\right) .$ 2) If $latex Fr\left(p\right) =$ $latex Fr\left( q\right) ,$ there is nothing to prove. Let $latex Fr\left(p\right) \neq Fr\left( q\right) .$ If $latex Fr\left( p\right) \cap Fr\left(q\right) =\varnothing ,$ then again there is nothing to prove. Suppose

(1) $latex Fr\left( p\right) \cap Fr\left( q\right) \neq \varnothing $ for some $latex p,q.$

Take any $latex r\in Fr\left( p\right) \cap Fr\left( q\right) $ ($latex r$ is a friend of both $latex p$ and $latex q).$ Then by Property 2 the inclusion $latex r\in Fr\left(q\right) $ can be reversed: $latex q\in Fr\left( r\right) .$ We get $latex r\in Fr\left(p\right) ,$ $latex q\in Fr\left( r\right) $ ($latex r$ is a friend of $latex p$ and $latex q$ is a friend of $latex r)$ and by Property 3 $latex q\in Fr\left( p\right) $ for any $latex q$ satisfying (1), which implies $latex Fr\left( q\right) \subset Fr(p)$. Similarly, $latex Fr\left( p\right) \subset Fr(q).$ Thus, we have the coincidence $latex Fr\left(p\right) =$ $latex Fr\left( q\right) .$

Now we can define the relation $latex \sim$ on the set of all people by saying that $latex p\sim q$ if $latex q\in Fr\left( p\right) .$ With this notation, assumptions a)-c) easily translate to the axioms:

Definition 2. The relation $latex \sim$ we just defined is an equivalence relation properties, that is, it satisfies the axioms:

Axiom 1. Reflexivity. For any $latex p,$ we have $latex p\sim p$ (follows from a)).

Axiom 2. Symmetry. $latex q\sim p$ implies $latex p\sim q$ (follows from b)).

Axiom 3. Transitivity. $latex q\sim p$ and $latex r\sim q$ implies $latex r\sim p$ (follows from c)).

On the picture: level sets define an equivalence relation: all points belonging to the same level set are declared to be equivalent to each other.

Level set