As discussed earlier, the sum of numbers of individual sets is greater than the number of elements in “

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 “

### Note:

**Union of three sets (Analytical method)**

We can achieve this result analytically as well. Here, we consider “A” as one set and “

Applying result for the union of two sets for “

Putting in the expression for “

At this stage, our task is to evaluate “

We can treat each of the terms in the small bracket on the right hand side of the above equation as a set. Applying relation obtained for the numbers in the union of two sets again, we have :

The last term in above equation is :

Hence,

Now, putting this expression in the expression of the numbers in the union involving three sets and rearranging terms, we have :

In the nutshell, we find that numbers of elements in the union, here, is equal to the sum of numbers in the individual sets, minus elements common to two sets taken at a time, plus elements common to all three sets.