When a set is formed with no elements it cannot be termed as nothing but a set with no elements in it. Such sets which contain no elements are addressed as empty set or null set.
Empty Set – Definition
Defining an empty set would be quite subtle and requires a thought. It is important to make out that a set is considered to be a collection of elements. The set is different from the elements it includes.
For instance consider {7}, which is a set that contains element 7. The set {7} is not a number. It is a set with number 7 as an element, while 7 is considered a number. An empty set, with that perspective isn’t nothing; rather, it is a set with no elements. It lets you think about sets as containers and the elements are things put into it. An empty container is however, analogous to empty set.
Notation and Terminology for an Empty Set
An empty set is supposed to be denoted by the ∅ symbol. As mentioned before, an empty set is also called a null set.
Properties of an Empty Set
Since we are considering only one empty set, it may be worthwhile to recognize what happens when set operations like intersection, union and complement are used along with empty set and a general set that is denoted by a variable say X. Mentioned below are a few factors regarding the same.
- The intersection of any considerable set with an empty set is the empty set. This is because there are actually no elements present in the empty set. Therefore the two sets hold no common elements. The same when written in symbols, would be, X ∩ ∅ = ∅.
- The union of any considerable set with an empty set would result in the set that we had discussed in the beginning. The reason for this is, there are no elements within an empty set. Therefore, we are not adding any new elements to the other set while a union is formed. In symbols, this is written as: X U ∅ = X.
- Complement of an empty set is a universal set for the instance that we have considered here. The reason for this is that the set of all elements which is absolutely not the part of the empty set would be determined as the set of all elements.
- An empty set is known to be the subset of any considerable set. The reason behind this is, we form subsets of a considered set X by either selecting or not selecting the elements from it. The subsets have an option to use no elements from the set. This is how an empty set is obtained.
The above-factors help in dealing with an empty set appropriately.
