Intersection (set Theory) - Nullary Intersection

Nullary Intersection

Note that in the previous section we excluded the case where M was the empty set (∅). The reason is as follows: The intersection of the collection M is defined as the set (see set-builder notation)

If M is empty there are no sets A in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), which according to standard (ZFC) set theory, does not exist.

A partial fix for this problem can be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption and becomes the identity element for this operation.