Final Exam
Ma 320
Carter
General Instructions. Write neat, complete, articulate solutions to the problems below.
Each problem is worth 25 points. Work problem #1 and #2, and choose four additional
problems to work from among problems 2 through 9. If you decide to work more than
four problems, mark the four that you wish to account for your grade on the exam. The
other solutions will contribute to extra credit. Any student who completes all nine of these
problems correctly will receive an A in the course. I have enjoyed being able to speak to you
about mathematics. I wish you success on this exam and in your subsequent mathematics
courses.
Never underestimate the deliciousness of a wedge of lettuce, slice of tomato, and a ramekin
of your favorite dressing.
1. Define the following terms.
(a) A relation from a set X to a set Y .
(b) The union of sets A and B.
(c) A function from a set X to a set Y .
(d) An injective function.
(e) A surjective function.
≡
2. Suppose that n is a non-negative integer. Thus n ∈ Z+ . Define a relation Z ←n Z by
the formula a ≡n b if and only if n|(b − a). That is, there is an integer q such that
(b − a) = nq.
(a) Show that ≡n is a reflexive relation.
(b) Show that ≡n is a symmetric relation.
(c) Show that ≡n is a transitive relation.
3. This question concerns the power set of a set X.
(a) What are the elements in the power set of the set A = {1, 2, 3, 4}?
(b) Prove (by induction) that if a set X has n-elements, then the power set P(X) has
2n elements.
4. State the negative of the expression: “∀ > 0, ∃δ > 0 such that |x − 2| < δ implies
that |f (x) − f (2)| < .”
5. Prove that if n is not prime, then 2n − 1 is not prime.
6. Prove that
2
2
2
1 + 2 + ··· + n =
n
X
k=1
1
k2 =
n(n + 1)(2n + 1)
6
7. Complete the square of y = Ax2 + Bx = C to prove the quadratic formula.
8. Consider the real valued function
y=
x−1
.
x+2
(a) What is the domain of this function?
(b) What is its range?
(c) Find the inverse function.
(d) Verify that this is the inverse.
9. Given that 0! = 1, and for 0 < n, define n! = n(n − 1)!,
(a) prove that
(n − 1)!
(n − 1)!
n!
=
+
.
k!(n − k)!
(k − 1)!(n − k)! (k)!(n − 1 − k)!
(b) Given that
n
k
=
n!
,
k!(n−k)!
prove that
n X
n n−k k
(x + y) =
x y .
k
k=0
n
2