Math / Discrete

QuestionUse the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. $\begin{array}{l} n(A)=30 \\ n(B)=50 \\ n(A \cup B)=70 \\ n(U)=200 \end{array}$ a. $n\left(A \cap B^{\prime}\right)=$ $\square$

Studdy Solution

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 $n(A \cap B)$ :
$70 = 80 - n(A \cap B)$ $n(A \cap B) = 80 - 70$ $n(A \cap B) = 10$

STEP 4

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:
$\boxed{20}$

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