Questions: Suppose n(A)=18, n(B)=23, and n(A ∪ B)=23. Use a Venn diagram to find n(A ∩ B).

Transcript text: Suppose $n(A)=18, n(B)=23$, and $n(A \cup B)=23$. Use a Venn diagram to find $n(A \cap B)$.

Solution

Solution Steps

Step 1: Given Information

We are provided with the following values:

$ n(A) = 18 $
$ n(B) = 23 $
$ n(A \cup B) = 23 $

Step 2: Apply the Inclusion-Exclusion Principle

Using the principle of inclusion-exclusion, we can express the number of elements in the union of two sets as: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Step 3: Rearrange the Formula

To find the number of elements in the intersection of sets A and B, we rearrange the formula: \[ n(A \cap B) = n(A) + n(B) - n(A \cup B) \]

Step 4: Substitute the Values

Substituting the known values into the rearranged formula gives: \[ n(A \cap B) = 18 + 23 - 23 \]

Step 5: Calculate the Intersection

Performing the calculation results in: \[ n(A \cap B) = 18 \]

Final Answer

$\boxed{n(A \cap B) = 18}$

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