Today, let's take a look at the concepts of three overlapping sets. Look at the diagram given below.

Set A = *a + d + f + k*

Set B = *b + d + e + k*

Set C = *c + f + e + k*

Elements belonging to exactly one set = *a + b + c*

Elements belonging to exactly two sets = *d + e + f*

Elements belonging to all three sets = *k*

Let’s try to understand what all information we can find from this diagram.

1. Which regions represent the elements that belong to at least two sets?

The regions representing at least two sets are those in which elements that belong to two sets as well as three sets fall.

At least two sets: *d + e + f + k*

2. Which regions represent the elements that belong to at least one set?

All elements falling in any of the circles represent at least one set. The only elements that do not belong are those in the white space of “None”

At least one set: *a + b + c + d + e + f + k* = Total – None

3. Which regions represent the elements that belong to at most 2 sets?

Elements falling in no set, in one set and in two sets will belong to at most two sets.

At most two sets: None *+ a + b + c + d + e + f* = Total – *k*

4. Which regions represent the elements that belong to only one set?

Elements falling in only one set are those in blue, yellow and red regions.

Only one set: *a + b + c*

So how do we get the total elements (*a + b + c + d + e + f + k*) ?

Total = Set A + Set B + Set C - (Set A and Set B) - (Set B and Set C) - (Set C and Set A) + (Set A and Set B and Set C)

(*a + d + f + k) + (b + d + e + k) + (c + f + e + k) - (d + k) - (f + k) - (e + k) + k which gives us *(*a + b + c + d + e + f + k*)

We need to add back k because we subtracted k three times and hence the k region got fully removed. But we do want it in our total.

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