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4) If

A

and

B

are subsets of the universal set

X

and $n(X)=50, n(A)=35, \...

Jul 25, 2024

Solution by Steps

step 1

Given:

n(X) = 50

n(A) = 35

n(B) = 20

, and

n(A' \cap B') = 5

. We need to find

n(A \cup B)

. Using the formula for the union of two sets:

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

step 2

To find

n(A \cap B)

, we use the fact that

n(A' \cap B') = n(X) - n(A \cup B)

. Since

n(A' \cap B') = 5

, we have

n(A \cup B) = 50 - 5 = 45

step 3

Now, we substitute

n(A \cup B) = 45

into the union formula:

45 = 35 + 20 - n(A \cap B)

. Solving for

n(A \cap B)

, we get

n(A \cap B) = 10

step 4

To find

n(A' \cap B)

, we use the formula:

n(A' \cap B) = n(B) - n(A \cap B)

. Substituting the values, we get

n(A' \cap B) = 20 - 10 = 10

step 5

To find

n(A \cap B')

, we use the formula:

n(A \cap B') = n(A) - n(A \cap B)

. Substituting the values, we get

n(A \cap B') = 35 - 10 = 25

Answer

n(A \cup B) = 45

n(A \cap B) = 10

n(A' \cap B) = 10

n(A \cap B') = 25

Key Concept

Set operations and their properties

Explanation

We used the properties of set operations and the given values to find the required quantities. The key formulas used were for the union and intersection of sets.