Elements in Three Sets: A, B, and C

How many elements are there in total?

Given three sets: a, b, and c. Set a has 14 elements, set b has 20 elements, and set c has 16 elements. The pair-wise intersections have 6 elements, and the three sets share 2 elements. What is the total number of elements in the three sets?

Answer:

There are a total of 24 elements in the three sets (a, b, and c). To find the total number of elements in the three sets (a, b, and c), we can use the principle of inclusion-exclusion.

Let's denote the total number of elements in each set as |a| = 14, |b| = 20, and |c| = 16. We also know that the pair-wise intersections have 6 elements each: |a ∩ b| = 6, |a ∩ c| = 6, and |b ∩ c| = 6. Additionally, all three sets share 2 elements: |a ∩ b ∩ c| = 2.

The principle of inclusion-exclusion states that to find the total number of elements in the union of the three sets, we need to sum the individual set sizes, subtract the sizes of the pair-wise intersections, and add back the size of the intersection of all three sets.

Using the formula for inclusion-exclusion, we have:

|a ∪ b ∪ c| = |a| + |b| + |c| - |a ∩ b| - |a ∩ c| - |b ∩ c| + |a ∩ b ∩ c|

Substituting the given values:

|a ∪ b ∪ c| = 14 + 20 + 16 - 6 - 6 - 6 + 2

Calculating this expression:

|a ∪ b ∪ c| = 40 - 18 + 2

|a ∪ b ∪ c| = 24

Therefore, there are a total of 24 elements in the three sets (a, b, and c).