Empty set

An empty set, as obviously indicates, is a set which is empty. In other words in an empty-set there are no elements. Sometimes it is also referred as null set. An empty set symbol can be any of the followings.      { } , (/),Ø. Of all these, the last symbol is very prominent.

It may be noted that it is not same as the Greek letter F. There is an interesting point in referring about empty-set. A set when empty is in a way common to all the sets that we think of. Therefore a set with no elements is better referred as ‘the empty set’, the word ‘the’ is more apt than the word ‘an’.

The null set, apart from the fact of having no elements has other properties too. The empty set can be a subset of any set. This may be a bit difficult to understand but a Venn diagram will be helpful to understand clearly. The union of the null set with any set A is the set A itself. It is obvious because we add only a 0 to the number of elements of A.

The intersection of the null set with any set A is again the null set. It is very clear that the intersection set cannot have any elements and it has to be empty. There can be only one subset of the null set which is the null set itself.

Now let us discuss about power set of the empty set. Before that let us see what a power set is. A power set of a set A is the set of all possible subsets of set A, including the null set. For example, if A is {a, b, c}, then all the possible subsets are { }, {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}. Therefore, the power set AP(A) is defined as {{ }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}. Thus we find there are 8 subsets for a set A having three elements.

That is, 23 subsets for 3 elements and in set language the power set P(A) in this case can be expressed as P(A) = 23.  Extending this concept in general for a set A having ‘n’ number of elements, P(Ø) = 2n. Therefore, in case of a null set the power set of the null set is P(A) = 20 = 1. That is the null set itself. In other words, P(Ø) = { }.