1. **State the problem:** We are given a table of animal counts by class and time of highest activity. We need to find the cardinal numbers of the sets: - $n(A \cup E)$ - $n(G \cup (D \cap F))$ Given: - $n(B \cap G) = 14$ 2. **Recall set operations and formulas:** - The union of two sets $X$ and $Y$ is $n(X \cup Y) = n(X) + n(Y) - n(X \cap Y)$. - The intersection $n(D \cap F)$ is the number of fish active in the afternoon. 3. **Find $n(A \cup E)$:** - $n(A)$ (Mammals total) = 164 - $n(E)$ (Morning total) = 269 - $n(A \cap E)$ is mammals active in the morning = 120 Using the union formula: $$ \begin{aligned} n(A \cup E) &= n(A) + n(E) - n(A \cap E) \\ &= 164 + 269 - 120 \\ &= 313 \end{aligned} $$ 4. **Find $n(G \cup (D \cap F))$:** - $n(G)$ (Evening total) = 98 - $n(D \cap F)$ is fish active in the afternoon = 29 - $n(G \cap (D \cap F))$ is fish active in both evening and afternoon, which is impossible (times do not overlap), so $n(G \cap (D \cap F)) = 0$ Using the union formula: $$ \begin{aligned} n(G \cup (D \cap F)) &= n(G) + n(D \cap F) - n(G \cap (D \cap F)) \\ &= 98 + 29 - 0 \\ &= 127 \end{aligned} $$ **Final answers:** - $n(A \cup E) = 313$ - $n(G \cup (D \cap F)) = 127$