And these of venn diagram

You're on a roll — let's solve these Venn diagram questions. Given: $n(U) = 100$ $n(A) = 40$ $n(B) = 30$ $n(A \cap B) = 10$ a) Find $n(A \cup B)$ Step 1: Use the formula for the union of two sets. $$ n(A \cup B) = n(A) + n(B) - n(A \cap B) $$ Step 2: Substitute the given values. $$ n(A \cup B) = 40 + 30 - 10 $$ $$ n(A \cup B) = 70 - 10 $$ $$ n(A \cup B) = 60 $$ The number of elements in $A \cup B$ is $\boxed{60}$. b) Find $n(A \text{ only})$ Step 1: To find elements in A only, subtract the intersection from $n(A)$. $$ n(A \text{ only}) = n(A) - n(A \cap B) $$ Step 2: Substitute the given values. $$ n(A \text{ only}) = 40 - 10 $$ $$ n(A \text{ only}) = 30 $$ The number of elements in A only is $\boxed{30}$. c) Find $n(B \text{ only})$ Step 1: To find elements in B only, subtract the intersection from $n(B)$. $$ n(B \text{ only}) = n(B) - n(A \cap B) $$ Step 2: Substitute the given values. $$ n(B \text{ only}) = 30 - 10 $$ $$ n(B \text{ only}) = 20 $$ The number of elements in B only is $\boxed{20}$. d) Find $n(A \cup B)'$ Step 1: The complement of $A \cup B$ is found by subtracting $n(A \cup B)$ from the total number of elements in the universal set $n(U)$. $$ n(A \cup B)' = n(U) - n(A \cup B) $$ Step 2: Substitute the values. We found $n(A \cup B) = 60$ in part (a). $$ n(A \cup B)' = 100 - 60 $$ $$ n(A \cup B)' = 40 $$ The number of elements in $(A \cup B)'$ is $\boxed{40}$. e) Find $n(A' \cap B')$ Step 1: Apply De Morgan's Law, which states that $A' \cap B' = (A \cup B)'$. $$ n(A' \cap B') = n((A \cup B)') $$ Step 2: This is the same value calculated in part (d). $$ n(A' \cap B') = 40 $$ The number of elements in $A' \cap B'$ is $\boxed{40}$. What's next?