1. **State the problem:** We are given the cardinalities of complements and unions of sets and need to find the number of elements in the set $n(A \cap B)$. 2. **Given:** - $n(A') = 70$ - $n(B') = 100$ - $n(A' \cup B') = 120$ - $n(U) = 160$ - $n(A \cap B') = 50$ - Find $n(A \cap B)$ 3. **Recall important formulas:** - $n(A) = n(U) - n(A')$ - $n(B) = n(U) - n(B')$ - $n(A' \cup B') = n(U) - n(A \cap B)$ (De Morgan's law) 4. **Calculate $n(A)$ and $n(B)$:** $$ n(A) = 160 - 70 = 90 $$ $$ n(B) = 160 - 100 = 60 $$ 5. **Use De Morgan's law to find $n(A \cap B)$:** $$ n(A' \cup B') = 120 = n(U) - n(A \cap B) = 160 - n(A \cap B) $$ Rearranging: $$ n(A \cap B) = 160 - 120 = 40 $$ 6. **Verify with given $n(A \cap B')$:** Since $n(A) = n(A \cap B) + n(A \cap B')$, then $$ 90 = n(A \cap B) + 50 $$ $$ n(A \cap B) = 90 - 50 = 40 $$ This matches our previous result. **Final answer:** $$ n(A \cap B) = 40 $$