Question

If \(n(A)=15, n(B)=20,\) and \(n(A \cap B)=10\) find \(n(A \cup B)\)

Short answer

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

Step-by-step solution

Identify Given Values

First, identify the values provided in the problem. Here, we are given that: - The number of elements in set A is 15: \(n(A) = 15\) - The number of elements in set B is 20: \(n(B) = 20\) - The number of elements in the intersection of sets A and B is 10: \(n(A \cap B) = 10\)

Use the Formula for Union of Two Sets

To find the number of elements in the union of sets A and B, use the formula: \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]This formula accounts for the fact that the intersection of the two sets is counted twice when adding the number of elements in each set.

Substitute Given Values into the Formula

Now, plug the known values into the formula:\[n(A \cup B) = 15 + 20 - 10 \]Simplify the expression to find the result.

Perform the Calculation

Calculate the result:\[n(A \cup B) = 15 + 20 - 10 = 25\]

Key concepts

Union of Sets

The union of two sets includes all the unique elements present in either one or both of the sets. When we talk about the union, we use the symbol \(\cup\). For instance, if we have two sets, \(A\) and \(B\), then the union of these sets \(A \cup B\) will include every element from both sets, but each element appears only once. This principle prevents double-counting of any shared elements.

Let's consider an example to make this clear:

• Set A = {1, 2, 3}
• Set B = {3, 4, 5}

The union of A and B, \(A \cup B\), will be {1, 2, 3, 4, 5}. Notice how the number 3 appears only once even though it is in both sets. This concept also applies when calculating the cardinality - or the number of elements - in the union of sets.

To find the number of elements in the union of two sets, say \(A\) and \(B\), we use the formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] This formula ensures that elements common to both sets are not double-counted by subtracting the number of elements in the intersection.

Intersection of Sets

The intersection of two sets includes only the elements common to both sets. The intersection symbol is \(\cap\). For example, if you have two sets A and B, their intersection \(A \cap B\) will include only those elements that are in both A and B.

Let's see this through an example:

• Set A = {1, 2, 3}
• Set B = {3, 4, 5}

Here, the intersection \(A \cap B\) is {3}, because 3 is the only element that exists in both sets.

When calculating the number of elements in the intersection of two sets, we count only the shared elements. This is crucial when using the union formula to ensure accurate results. In the given exercise, set A and set B intersect at 10 elements, denoted as \(n(A \cap B) = 10\).

Recognizing the intersection's significance ensures proper comprehension when working with combined sets and avoids duplication in the union formula.

Cardinality of Sets

Cardinality refers to the number of elements in a set. It's a measure of the set's 'size.' If set A has 15 elements, we express this as \(n(A) = 15\). Likewise, if set B has 20 elements, we express it as \(n(B) = 20\).

In set theory, understanding cardinality helps us determine the total number of unique elements when sets combine. Using our earlier concepts, if we want to find the cardinality of the union of two sets, A and B, we need to account for duplicates by using the intersection information.

Given the values:

• \(n(A) = 15\)
• \(n(B) = 20\)
• \(n(A \cap B) = 10\)

The cardinality of the union of the sets, \(n(A \cup B)\), is computed using: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
This calculation becomes: \[ 15 + 20 - 10 = 25 \] Thus, \(n(A \cup B) = 25\), meaning the union of sets A and B has 25 unique elements. Efficiently understanding and applying these concepts is essential in many mathematical and practical contexts.