Question

If \(n(A)=30, n(B)=40,\) and \(n(A \cup B)=45\) find \(n(A \cap B)\)

Short answer

The number of elements in the intersection of sets A and B is 25.

Step-by-step solution

Understand the given information

Identify the values provided in the problem. Here, we are given the following: - Number of elements in set A, denoted as \(n(A) = 30\)- Number of elements in set B, denoted as \(n(B) = 40\)- Number of elements in the union of sets A and B, denoted as \(n(A \, \cup \, B) = 45\)

Recall the formula for the union of two sets

Use the principle of inclusion and exclusion for the union of two sets. The formula is:\[ n(A \, \cup \, B) = n(A) + n(B) - n(A \, \cap \, B) \]Where \( n(A \, \cap \, B) \) represents the number of elements in the intersection of sets A and B.

Substitute the known values into the formula

Plug the known values into the formula from Step 2:\[ 45 = 30 + 40 - n(A \, \cap \, B) \]

Solve for the intersection

Isolate \(n(A \, \cap \, B)\) on one side of the equation:\[ 45 = 30 + 40 - n(A \, \cap \, B) \]Simplify the right-hand side:\[ 45 = 70 - n(A \, \cap \, B) \]Subtract 70 from both sides:\[ 45 - 70 = -n(A \, \cap \, B) \]Simplify:\[ -25 = -n(A \, \cap \, B) \]Multiply both sides by -1:\[ 25 = n(A \, \cap \, B) \]

Key concepts

union of sets

In set theory, the union of two sets A and B, denoted as \(A \, \bigcup \, B\), is a set containing all elements from both A and B. For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \, \bigcup \, B = \{1, 2, 3, 4, 5\}\). The union operation combines all unique elements from both sets.
To determine the total number of elements in \(A \, \bigcup \, B\), we can use the principle of inclusion and exclusion (PIE). The formula for the number of elements in the union is:
\[n(A \, \bigcup \, B) = n(A) + n(B) - n(A \, \bigcap \, B)\]
This formula avoids counting the elements in the intersection \(A \, \bigcap \, B\) twice. Understanding the union of sets is crucial for solving problems involving multiple sets.

intersection of sets

The intersection of two sets A and B, denoted as \(A \, \bigcap \, B\), is a set of elements that are present in both A and B. For instance, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \, \bigcap \, B = \{3\}\).
Finding the intersection is key to many set operation problems. To use the principle of inclusion and exclusion, it's important to be able to calculate \(n(A \, \bigcap \, B)\), the number of elements in the intersection of sets A and B.
In our example problem, we start with the provided values:

• \(n(A)=30\)
• \(n(B)=40\)
• \(n(A \, \bigcup \, B)=45\)

Using the formula: \[n(A \, \bigcap \, B) = n(A) + n(B) - n(A \, \bigcup \, B)\]We substitute and solve:
\[n(A \, \bigcap \, B) = 30 + 40 - 45 = 25\]
This gives us the number of elements in the intersection of A and B.

principle of inclusion and exclusion

The principle of inclusion and exclusion (PIE) is a fundamental concept used to accurately count the number of elements in the union of multiple sets. PIE helps to avoid the over-counting of elements that may appear in more than one set.
For two sets A and B, the PIE formula is:
\[n(A \, \bigcup \, B) = n(A) + n(B) - n(A \, \bigcap \, B)\]
This formula takes into account the total elements of A and B and subtracts the common elements counted twice when summing \(n(A)\) and \(n(B)\).
In the provided exercise, to find the intersection \(n(A \, \bigcap \, B)\), we used PIE and substituted known values:

• \(n(A)=30\)
• \(n(B)=40\)
• \(n(A \, \bigcup \, B)=45\)

Applying the formula:
\[n(A \, \bigcup \, B) = n(A) + n(B) - n(A \, \bigcap \, B)\]Solving for the intersection:
\[45 = 30 + 40 - n(A \, \bigcap \, B)\]
\[n(A \, \bigcap \, B) = 25\]
Thus, PIE provided an accurate way to find the number of elements common to both sets.