We are given the following information:

• $n(A') = 70$
• $n(B') = 100$
• $n(A' \cup B') = 120$
• $n(U) = 150$

We are asked to find $n(A \cap B')$.

Step 1: Use the formula for set complements

We know that:

$n(A' \cup B') = n(U) - n(A \cap B)$

This comes from the fact that the complement of $A' \cup B'$ is $A \cap B$. Rearranging the formula to find $n(A \cap B)$:

$n(A \cap B) = n(U) - n(A' \cup B')$

Substitute the known values:

$n(A \cap B) = 150 - 120 = 30$

Step 2: Apply the inclusion-exclusion principle

The inclusion-exclusion principle for sets states that:

$n(A' \cup B') = n(A') + n(B') - n(A' \cap B')$

Substitute the known values:

$120 = 70 + 100 - n(A' \cap B')$

Solve for $n(A' \cap B')$:

$120 = 170 - n(A' \cap B')$ $n(A' \cap B') = 170 - 120 = 50$

Step 3: Find $n(A \cap B')$

We know that $A' \cap B'$ and $A \cap B'$ are disjoint sets. Also, the union of these two sets is $B'$, so:

$n(B') = n(A' \cap B') + n(A \cap B')$

Substitute the known values:

$100 = 50 + n(A \cap B')$

Solve for $n(A \cap B')$:

$n(A \cap B') = 100 - 50 = 50$

Thus, $n(A \cap B') = 50$.

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Do you have any further questions or need more details?

Here are 5 related questions to expand on the topic:

1. What is the difference between the complement of a set and the intersection of two sets?
2. How does the inclusion-exclusion principle apply when dealing with three sets?
3. How would you find $n(A \cup B)$ using the given information?
4. Can you calculate $n(B \cap A')$ from the given data?
5. How do you calculate the number of elements in the union of three sets?

Tip: Always keep the set operations' relationships in mind, like De Morgan’s Laws, when working with complements and unions.