"Having the same age" is an equivalence relation between people.

• It is reflexive: Bob is always the same age as himself

• It is symmetric: if Bob is the same age as Sally, then Sally is the same age as Bob

• It is transitive: If Bob is the same age as Sally, and Sally is the same age as Fred, then Bob is the same age as Fred.

using symbols:

Bob ~ Bob

Bob ~ Sally ⇒ Sally ~ Bob

Bob ~ Sally and Sally ~ Fred ⇒ Bob ~ Fred 

"⇒" means "the statement on the left implies the statement on the right." When people in this thread write =>, <=, and <=> they mean ⇒, ⇐, and ⇔

An "equivalence class" is the set of all items that obey the equivalence relation with each other. So, "being 25 years old" is an equivalence class containing every person who is 25 years old. Those people might be different in every other way, but they are equivalent in that specific regard.

In their proof earlier, @[email protected] defined two equivalence classes. Instead of "people who are 25 years old," the classes were "infinite sequences that converge to 1" and "infinite sequences that converge to 0.999...." They showed that these are the same class.