Question

a company has 7 job position openings: y1, . . . , y7. soon they receive applications from 10 job applicants x1, . . . , x10. the set of positions each applicant is qualified for is given, respectively, by {y1, y2, y6}, {y2, y6, y7}, {y3, y4}, {y1, y5}, {y6, y7}, {y3}, {y2, y3}, {y1, y3}, {y1}, and {y5}. what is the largest number of positions that can be filled by qualified applicants? justify your answer.

Answer 1

The maximum number of positions that can be filled by qualified applicants is 7. Each job position is covered by at least one applicant, ensuring a comprehensive filling of all available positions.

To maximize the number of positions filled, let's consider the qualifications of each job applicant for the available positions:

1.
`\( y1 \)` is qualified by applicants {x1, x4, x8, x9}.

2.
`\( y2 \)` is qualified by applicants {x1, x2, x7}.

3.
`\( y3 \)` is qualified by applicants {x3, x6, x7, x8, x9}.

4.
`\( y4 \)` is qualified by applicant {x3}.

5.
`\( y5 \)` is qualified by applicants {x4, x10}.

6.
`\( y6 \)` is qualified by applicants {x1, x2, x5}.

7.
`\( y7 \)` is qualified by applicants {x2, x5}.

Now, let's determine the largest number of positions that can be filled with qualified applicants:

- y1 : Filled by {x1}.

- y2 : Filled by {x2}.

- y3 : Filled by {x3}.

- y4 : Filled by {x3}.

- y5: Filled by {x4}.

- y6 : Filled by {x5}.

- y7 : Filled by {x2}.

Hence, the largest number of positions that can be filled is 7, with each position covered by at least one qualified applicant, ensuring all positions are filled.