1. **State the problem:** Given a universal set $U$ with $n(U) = 119$, and two subsets $A$ and $B$ with $n(A) = 70$, $n(B) = 48$, and $n(A \cup B) = 103$, find the number of elements in the intersection $n(A \cap B)$. 2. **Recall the formula for union of two sets:** $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ This formula accounts for the overlap counted twice when adding $n(A)$ and $n(B)$. 3. **Substitute the known values:** $$103 = 70 + 48 - n(A \cap B)$$ 4. **Simplify the equation:** $$103 = 118 - n(A \cap B)$$ 5. **Isolate $n(A \cap B)$:** $$n(A \cap B) = 118 - 103$$ 6. **Calculate the intersection:** $$n(A \cap B) = 15$$ 7. **Interpretation:** There are 15 elements common to both sets $A$ and $B$. --- **Venn Diagram values:** - Elements only in $A$ (not in $B$): $n(A) - n(A \cap B) = 70 - 15 = 55$ - Elements only in $B$ (not in $A$): $n(B) - n(A \cap B) = 48 - 15 = 33$ - Elements in both $A$ and $B$: $15$ - Elements outside both $A$ and $B$ in $U$: $n(U) - n(A \cup B) = 119 - 103 = 16$