1. **State the problem:** A company needs 30 programmers for system programming and 40 for application programming. They appoint 55 programmers in total. We need to find how many programmers do both jobs, only system programming, and only application programming. 2. **Use the formula for union of two sets:** $$|S \cup A| = |S| + |A| - |S \cap A|$$ where $|S|$ is the number of system programmers, $|A|$ is the number of application programmers, and $|S \cap A|$ is the number who do both. 3. **Plug in the known values:** $$55 = 30 + 40 - |S \cap A|$$ 4. **Solve for $|S \cap A|$:** $$|S \cap A| = 30 + 40 - 55 = 70 - 55 = 15$$ 5. **Find programmers who do only system programming:** $$|S| - |S \cap A| = 30 - 15 = 15$$ 6. **Find programmers who do only application programming:** $$|A| - |S \cap A| = 40 - 15 = 25$$ **Final answers:** - Programmers doing both jobs: 15 - Programmers doing only system programming: 15 - Programmers doing only application programming: 25