1. **State the problem:** We have a universal set $S = \{1, 2, 3, \ldots, 20\}$ and two subsets: Set $A = \{1, 4, 5, 9, 10, 11, 12, 13, 14, 16, 17, 19\}$ Set $B = \{5, 6, 9, 10, 11, 12, 13, 14, 18, 19\}$ We need to find: - The number of elements in the union $A \cup B$ - The number of elements in the intersection $A \cap B$ 2. **Recall formulas and rules:** - The union of two sets $A$ and $B$ is $A \cup B = \{x : x \in A \text{ or } x \in B\}$. - The intersection of two sets $A$ and $B$ is $A \cap B = \{x : x \in A \text{ and } x \in B\}$. - The number of elements in the union is given by: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ 3. **Find $n(A)$ and $n(B)$:** - Count elements in $A$: $$n(A) = 12$$ - Count elements in $B$: $$n(B) = 10$$ 4. **Find $A \cap B$ (common elements):** Elements in both $A$ and $B$ are: $$A \cap B = \{5, 9, 10, 11, 12, 13, 14, 19\}$$ Count them: $$n(A \cap B) = 8$$ 5. **Calculate $n(A \cup B)$ using the formula:** $$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 12 + 10 - 8$$ Show cancellation step: $$n(A \cup B) = \cancel{12} + \cancel{10} - 8 = 14$$ 6. **Final answers:** - Number of elements in $A \cup B$ is $14$. - Number of elements in $A \cap B$ is $8$. These results can be visualized with a Venn diagram showing the overlap of the two sets.