Question

Let a be the set of students who live within one mile of school and let b be the set of students who walk to classes. Describe the students in each of these sets. What is a intersection b? What is a union b? What is a - b? What is b - a?

1) The students who live within one mile of school and walk to classes
2) The students who live within one mile of school but do not walk to classes
3) The students who do not live within one mile of school but walk to classes
4) The students who do not live within one mile of school and do not walk to classes

Answer 1

The students who live within one mile of school and walk to classes can be found by finding the intersection of sets a and b. The students who live within one mile of school but do not walk to classes can be found by finding the set difference of a and b. The students who do not live within one mile of school but walk to classes can be found by finding the set difference of b and a.

Step-by-step explanation:

The set a represents the students who live within one mile of school. The set b represents the students who walk to classes. To describe the students in each set:

• a) The students who live within one mile of school and walk to classes would be the intersection of sets a and b.
• b) The students who live within one mile of school but do not walk to classes would be the set difference of a and b (a - b).
• c) The students who do not live within one mile of school but walk to classes would be the set difference of b and a (b - a).
• d) The students who do not live within one mile of school and do not walk to classes would be the complement of the union of sets a and b.

To find the intersection of sets a and b, we need to find the students who live within one mile of school and also walk to classes. This can be found by finding the common elements in sets a and b. The union of sets a and b would give us all the students who either live within one mile of school or walk to classes. Set difference a - b would give us the students who live within one mile of school but do not walk to classes, while set difference b - a would give us the students who do not live within one mile of school but walk to classes.

Answer 2

Set a includes students who live within one mile of school, and b includes those who walk to classes. The intersection a ∩ b is for students meeting both criteria, whereas the differences a - b and b - a represent students in one set but not the other. Union a ∪ b includes any student who meets at least one of the criteria.

Step-by-step explanation:

To answer the question, we need to understand the concepts of sets and their operations, such as intersection, union, and difference.

• Set a denotes the students who live within one mile of the school.
• Set b represents the students who walk to classes.
• The intersection of sets a and b, denoted a ∩ b, refers to students who belong to both sets - these are the students who live within one mile of school and walk to classes (1).
• The union of sets a and b, denoted a ∪ b, comprises all students who live within one mile of the school or walk to classes, including those who do both.
• The difference between sets a and b, denoted a - b, is the set of students who live within one mile of the school but do not walk to classes (2).
• The difference b - a consists of students who walk to classes but do not live within one mile of school (3).
• Those who neither live within one mile of the school nor walk to classes are not a part of any of these sets and thus refer to an outside group (4).