Math Snap

STEP 1

1. $n(A)$ represents the number of elements in set $A$.
2. $n(B)$ represents the number of elements in set $B$.
3. $n(A \cup B)$ represents the number of elements in the union of sets $A$ and $B$.
4. $n(U)$ represents the total number of elements in the universal set $U$.
5. $n\left(A \cap B^{\prime}\right)$ represents the number of elements in set $A$ that are not in set $B$.

STEP 2

1. Use the principle of inclusion-exclusion to find $n(A \cap B)$.
2. Calculate $n\left(A \cap B^{\prime}\right)$.

STEP 3

Use the principle of inclusion-exclusion to find $n(A \cap B)$:
$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ Substitute the given values:
$$70 = 30 + 50 - n(A \cap B)$$ Solve for $n(A \cap B)$:
$$70 = 80 - n(A \cap B)$$ $$n(A \cap B) = 80 - 70$$ $$n(A \cap B) = 10$$

Calculate $n\left(A \cap B^{\prime}\right)$:
$$n\left(A \cap B^{\prime}\right) = n(A) - n(A \cap B)$$ Substitute the known values:
$$n\left(A \cap B^{\prime}\right) = 30 - 10$$ $$n\left(A \cap B^{\prime}\right) = 20$$ The number of elements in $A \cap B^{\prime}$ is:
$$\boxed{20}$$