1. **State the problem:** We are given the number of people who own a cat but not a dog, own a dog but not a cat, and the total number of people who own a cat or a dog (or both). We want to find how many people own a dog and how many own both a cat and a dog. 2. **Given data:** - Own a cat but not a dog: 26 - Own a dog but not a cat: 145 - Own a cat or a dog (or both): 171 3. **Formula used:** For two sets $A$ (cat owners) and $B$ (dog owners), the union is: $$|A \cup B| = |A| + |B| - |A \cap B|$$ 4. **Define variables:** Let $x = |A \cap B|$ be the number of people who own both a cat and a dog. 5. **Express total owners:** We know: $$|A \cup B| = 171$$ $$|A| = 26 + x$$ (cat owners include those who own only cats plus those who own both) $$|B| = 145 + x$$ (dog owners include those who own only dogs plus those who own both) 6. **Apply the union formula:** $$171 = (26 + x) + (145 + x) - x$$ Simplify: $$171 = 26 + 145 + x$$ $$171 = 171 + x$$ 7. **Solve for $x$:** $$171 = 171 + x \implies x = 0$$ 8. **Interpretation:** - Number of people who own both a cat and a dog is $0$. - Number of people who own a dog is: $$|B| = 145 + 0 = 145$$ **Final answers:** - People who own a dog: $145$ - People who own both a cat and a dog: $0$