Venn Diagram 4E2194

1. **State the problem:** We are given the total number of elements in the universal set $n(U) = 150$, the number of elements in set $A$ as $n(A) = 57$, the number of elements in set $B$ as $n(B) = 90$, and the number of elements not in the union of $A$ and $B$ as $n((A \cup B)^c) = 14$. We need to analyze this using a Venn diagram. 2. **Recall the formula for the union of two sets:** $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ 3. **Calculate $n(A \cup B)$ using the complement:** Since $n((A \cup B)^c) = 14$, the number of elements in the union is: $$n(A \cup B) = n(U) - n((A \cup B)^c) = 150 - 14 = 136$$ 4. **Find the intersection $n(A \cap B)$:** Using the union formula: $$136 = 57 + 90 - n(A \cap B)$$ $$n(A \cap B) = 57 + 90 - 136 = 147 - 136 = 11$$ 5. **Summary:** - Total elements $n(U) = 150$ - Elements in $A$: $57$ - Elements in $B$: $90$ - Elements in $A \cap B$: $11$ - Elements in $A \cup B$: $136$ - Elements outside $A \cup B$: $14$ This information can be represented in a Venn diagram showing the overlap of $A$ and $B$ with $11$ elements in the intersection, $46$ elements only in $A$ ($57 - 11$), $79$ elements only in $B$ ($90 - 11$), and $14$ elements outside both sets.