1.6 Working with three sets

Working with three sets is similar as working with two sets. The underlying characteristics of set operations are same. The union and intersection of sets are carried out with three sets - one after another. Notably, union and intersection operations are commutative. This allows us to extend these set operations to third set in any sequence. Venn’s diagram enables us to visualize resulting set. In this module, we shall first formulate expression for the numbers in the union of three sets. Subsequently, we shall apply the formulation to real time analysis - using both graphical (Venn diagram) and analytical methods.

The union, involving three sets, can be considered in terms of union of a set with "union of other two sets". In that sense, union of three sets represent elements which belong to either of three sets. Here, we want to find the expression of numbers of elements in the union set, which is represented as :

$$n\left(A\cup B\cup C\right)$$

Before, we work on the expansion of this term, let us first find out what does the term " $A\cup B\cup C$ " represent on Venn’s diagram? The figure below shows the representation of this term :

The set shown in the figure above consists of following class of elements :

• The elements, which are exclusive of sets “A”, “B” and “C” respectively.
• The elements, which are common to a pair of two sets at a time.
• The elements, which are common to all three sets.

In the figure below, the common areas between a pair of two sets are marked “1”, “2” and “3”. The common area among all three sets is marked “4”.

Union of three sets

As discussed earlier, the sum of numbers of individual sets is greater than the number of elements in “ $A\cup B\cup C$ ”, unless the sets are disjoint sets. It is imperative that we account for the repetition of common elements. Proceeding as in the case of union of two sets, we deduct the intersections between each pair of sets as :

$$n\left(A\cup B\cup C\right)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n\left(A\cap B\right)-n\left(A\cap C\right)-n\left(B\cap C\right)$$

In this manner, we account for common elements between two sets. However, we have deducted elements "common to all three sets" in this process – three times. On the other hand, the elements "common to all three sets" are present in the numbers of each of the individual sets - in total three times as there are three sets. Ultimately, we find that we have not counted the elements common to all sets at all. It means that we need to account for the elements common to all three sets. In order to add this number, we first need to know – what does this common area (marked 4) represent symbolically?

In the earlier module, we have seen that the area marked “4” is represented by “ $A\cap B\cap C$ ”. Hence, the correct expansion for the numbers of elements in the union, involving three set, is :

$$\Rightarrow n\left(A\cup B\cup C\right)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n\left(A\cap B\right)-n\left(A\cap C\right)-n\left(B\cap C\right)+n\left(A\cap B\cap C\right)$$

Union of three sets (analytical method)

We can achieve this result analytically as well. Here, we consider “A” as one set and “ $\left(B\cup C\right)$ “ as other set. Then, we apply the relation, which has been obtained for the numbers in the union of two sets as :