Interpretation
What the axiom is really saying is that, given two sets A and B, we can find a set C whose members are precisely A and B. We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is:
- Any two sets have a pair.
{A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.
The axiom of pairing also allows for the definition of ordered pairs. For any sets and, the ordered pair is defined by the following:
Note that this definition satisfies the condition
Ordered n-tuples can be defined recursively as follows:
Read more about this topic: Axiom Of Pairing