Singleton definition from Indicator function

[mkphysics]

The attached image is from Wikipedia under the "Singleton" article.

What reason is the first sentence beginning "Let S be a class..." necessary to the definition?

I am confused by the words "defined by".

I think I understand the second sentence:

Take the second sentence beginning "Then S is called...":
Assume S={p,q} where p≠q, then S is not a singleton because there exists some y ϵ S for some x ϵ S such that b(p) = (p≠q). For example if y=p:
b(p) = (p=p)
b(p) = (p≠q) <-breaks the singleton logic

 

[Valkarie]

The first sentence beginning "Let S be a class..." is necessary to the definition because it sets up the parameters for the class in order to determine whether or not it is a singleton. It defines the class by specifying its elements and stating the conditions necessary for it to be a singleton. Without this sentence, there would be no way to know if the class is a singleton or not.