Understanding Set Intersection Using the Principle of Inclusion-Exclusion

In set theory, the intersection of two sets is a fundamental concept. The principle of inclusion-exclusion is a powerful tool that helps us determine the size of the union of sets. This principle can also be used to find the intersection of sets. Let's explore how to find n(A ∩ B) when given the sizes of sets A and B and their union.

Solving for n(A ∩ B) Using the Given Information

Given the principle of inclusion-exclusion for two sets, we have the formula:

[ n(A cup B) n(A) n(B) - n(A cap B) ]

Let's apply this formula to solve for n(A ∩ B). The given values are:

n(A) 30 n(B) 20 n(A ∪ B) 36

Substitute these values into the formula:

[ 36 30 20 - n(A cap B) ]

Simplify the equation:

[ 36 50 - n(A cap B) ]

Rearrange to solve for n(A ∩ B):

[ n(A cap B) 50 - 36 ][ n(A cap B) 14 ]

Therefore, the intersection of set A and set B, denoted as n(A ∩ B), is 14.

Internalizing the Concept

The process can be broken down as follows:

[ n(A cup B) n(A) n(B) - n(A cap B) ]

Substitute the values:

[ 36 30 20 - n(A cap B) ]

Further simplification results in:

[ 36 50 - n(A cap B) ]

Solving for n(A ∩ B) gives us:

[ n(A cap B) 50 - 36 ][ n(A cap B) 14 ]

Another format of the equation:

[ n(A cup B) n(A) n(B) - n(A cap B) ][ 36 30 20 - n(A cap B) ][ 36 50 - n(A cap B) ][ n(A cap B) 50 - 36 ][ n(A cap B) 14 ]

Both methods confirm that n(A ∩ B) 14.

Interpreting the Results

The value of n(A ∩ B) can range from 0 to the smaller of n(A) and n(B). Here are a few scenarios:

If A and B are disjoint sets, n(A ∩ B) is 0. This means sets A and B have no elements in common.If A is a subset of B, n(A ∩ B) is equal to n(A). This means all elements of A are also in B.If A intersects B partially, n(A ∩ B) can take any value between 0 and 20 (the smaller of n(A) and n(B)).

Given the context, 14 falls within the expected range, indicating that there are 14 elements common to both sets A and B.

Conclusion

Using the principle of inclusion-exclusion, we successfully calculated the intersection of sets A and B. The value of n(A ∩ B) is 14. This method can be applied to various problems involving set operations, making it a valuable tool in set theory and discrete mathematics.