How Universal Set and Complement Set are Related to Each Other?

Let us start with the definition of universal set and complement set in much more detail. Universal Set can be defined as a particular set that comprises all the elements of the associated sets and the elements in the Universal Set are not repeated on the other hand complement sets consist of all such elements which are present in the Universal Set but don’t make it in those particular sets.

Universal Set

Here I will try to explain the universal set to the best of my ability. If you consider all the sets in the mathematical world, then the superset of all those sets will be the universal set. What that basically means is, The universal set consists of all the elements of all the sets in the mathematical world. The Universal Set can be denoted by E or U. One more way of describing The Universal Set is that all the sets in this world are the subsets of The Universal Set.

Now we will try to explain an example of a Universal Set. Consider three sets X, Y and Z. Let us assume that X= {7, 12, 45, 65}, Y= {117, 43, 1} and Z= {0}. Note: Z is not an empty set; it consists of the element 0. So the Universal Set of X, Y, and Z will be the set that consists of all the elements in X, Y, and Z. So the Universal Set (U) of the sets X, Y, and Z will be given as, 

U= {7, 12, 45, 65, 117, 43, 1, 0}.

Please do note that in the Universal Set, the elements are not repeated. That basically means that if the same element is included in two or more sets, then that element will be included only one time in the Universal Set of those sets.

Now we will see a solved problem related to the Universal Set.

1. Find the Universal Set of the sets A={5,6,7,8}, B={8,9,10,11} and C={11,12,13,14}.

Soln. 

We already know that the Universal Set of a particular group of sets is the superset of those particular sets.

So the Universal Set of A, B and C is:-

U={5,6,7,8,9,10,11,12,13,14}.

One interesting fact about the Universal Set is that the complement set of the universal set is the empty set which is denoted by φ. The Empty Set can also be represented as {}. An empty set can be described as a set that has no elements or zero elements. Please note that The Empty Set is very different from a set that has 0 as an element.

Now we will move on to the complement set. 

Complement Set

Now I will try to explain the complement set. For that, we will first consider the universal set and a subset of the universal set. Let the subset be called A. So the complement set of A will consists of all those elements which are present in the universal set but are not present in A. The complement set of A will be denoted as A’. We can also describe the complement set by saying that, the complement set of a set A is the universal set minus the set A. Now we will look at an example of the complement set.

Consider a universal set U={4,7,8,3,15,72} and a particular set A={7,8,3,72}. Find the Complement Set of A.

We know that the Complement Set of a particular set consists of all those elements which are present in the Universal Set but not in the set.

So A’= {4, 15}

Now we will see a question related to the Complement Set.

1. Consider a Universal Set U= {5, 3, 7, 13, 9, 41} and a related set A= {3, 13, 9}. Find The Complement Set of A.

Soln.

As we already know the Complement of a particular set is the set of all elements which are present in a Universal Set but not present in the related set.

The Complement of the Set A is given by:-

A’= {5, 7, 41}

Now we will see some properties related to the Complement Set.

The first property tries to explain that since a particular set is the subset of its universal set, its complement will also be the subset of the universal set.

The second property states that if we try to take an intersection of a particular set and its complement set, that intersection set would be the empty set.

Random Properties

1. If try to take the complement of the complement set of a particular set, you will get that particular set as the solution.
2. The complement set of the universal set is the empty set and the complement set of the empty set is the universal set.

Conclusion

Here, in this article, we first described the Universal Set and the Complement Set. We have also seen how both the universal set and the complement set are related to each other. Finally, we looked at some random properties related to these sets.