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 $n(B \cup E)$ and $n(G \cup (A \cap F))$, given $n(B \cap G) = 25$. 2. **Recall set operations formulas:** - For any two sets $X$ and $Y$, the cardinality of their union is: $$n(X \cup Y) = n(X) + n(Y) - n(X \cap Y)$$ 3. **Identify given values from the table:** - $n(B) = 140$ (total birds) - $n(E) = 310$ (total morning observations) - $n(G) = 127$ (total evening observations) - $n(A \cap F) = 57$ (mammals in afternoon) - $n(B \cap G) = 25$ (birds in evening) 4. **Calculate $n(B \cup E)$:** - We need $n(B \cap E)$ (birds in morning). From the table, birds in morning $= 59$. - Using the formula: $$n(B \cup E) = n(B) + n(E) - n(B \cap E) = 140 + 310 - 59 = 390$$ 5. **Calculate $n(G \cup (A \cap F))$:** - $n(G) = 127$ - $n(A \cap F) = 57$ - We need $n(G \cap (A \cap F))$, which is the intersection of evening observations and mammals in afternoon. Since these are disjoint times, this intersection is 0. - Using the formula: $$n(G \cup (A \cap F)) = n(G) + n(A \cap F) - n(G \cap (A \cap F)) = 127 + 57 - 0 = 184$$ **Final answers:** - $n(B \cup E) = 390$ - $n(G \cup (A \cap F)) = 184$