Exercises 13To prove "if a, then b" and "a if and only if b" results, use one of the two forms:Form 1: To prove "if a, then b," assume a and derive b.
Form 2: To prove "a if and only if b," prove "if a, then b," and then prove "if b,
then a."
In a proof of an if and only if statement, a proof of "if a, then b" is usually labeled
(=#), whereas a proof of "if b, then a" is usually labeled (4=). The if and only if
statement is often written using •.rnExercises
- Let X be the set of all students at a university. Let A be the set of students who are first-
year students, B the set of students who are second-year students, C the set of students
who are in a discrete mathematics course, D the set of students who are international
relations majors, E the set of students who went to a concert on Monday night, and F
the set of students who studied until 2 AM on Tuesday. Express in set theoretic notation
the following sets of students:
(a) All second-year students in the discrete mathematics course.
Sample Solution. {X E X : x E B and x c C}.
(b) All first-year students who studied until 2 AM on Tuesday.
(c) All students who are international relations majors and went to the concert on
Monday night.
(d) All students who studied until 2 AM on Tuesday, are second-year students, and are
not international relations majors.
(e) All first- and second-year students who did not go to the concert on Monday night
but are international relations majors.
(f) All students who are first-year international relations majors or who studied until
2 AM on Tuesday.
(g) All students who are first- or second-year students who went to a concert on Mon-
day night.
(h) All first-year students who are international relations majors or went to a concert
on Monday night. - Find at least two different ways to fill in the ellipses in the set descriptions given.
For example, {2, 4 ... , 121 could be written either {2n : 1 < n < 6 and n G N} or
{n+ I :n E {1,3,5,7, 111}.
(a) {1, 3,..., 311
(b) 11, 2,..., 26)
(c) {2, 5,..., 321