Empty Set or Null Set

In set theory the concept of an empty set or null set is very important and interesting. Its definition is as follows:  “a set which contains no elements is called as empty set or null set”, and it is sometimes known as void set or vacuous set. It is usually denoted by \phi ; this Greek symbol is known as phi.

This definition of empty seems to be opposing the definition of a set, but this apparent contradiction makes sense when we understand the meanings of well defined objects. By well defined objects we mean objects which can be considered as the members of the set under consideration.

For example, a bottle is not a well defined object for a jewelry set. Similarly, a chair is not a well defined object for a tea set. But if we consider the set of necessary things of a house, then both of the above mentioned things are well defined objects for such a set.

Thus, if an object does not exist then it is a well defined object for an empty set.

For example,

  1. Odd numbers which are divisible by 2 are well defined objects for an empty set. A hen with horns is also a well defined object for an empty set.
  2. The positive integers which are less than -1 are also well defined objects for an empty set. If {\mathbb{Z}^ + } denotes the set of all positive integers, then the last set may be represented as

\phi = \left\{ {x:x \in {\mathbb{Z}^ + } \wedge x < - 1} \right\}

  1. Let A be the set all those people whose height is greater than 50 feet. According to the present statistical analysis of the world a person exists whose height is greater than 50 feet, so A is null set.
  2. LetB = \left\{ {x:{x^2} = 4,\,is\,odd} \right\}. In this situation B is the empty set.