Math 309
Quiz 1 Solutions
Summer 2014
Drew Armstrong
1. Let n be a positive whole number. Find a closed form for the following sum:
1 + 2 + 3 + + n =
1
1
n(n + 1)
= n2 + n
2
2
2
2. Let n be a positive whole number. Find a closed form for the following sum

Math 309
Homework 5
Summer 2014
Drew Armstrong
The purpose of this homework is to let you practice the technique of induction. Each proof
should take up a good amount of space. Dont try to shrink it down to one paragraph. Make
sure that the logical struct

Math 309
Homework 1 Solutions
Summer 2014
Drew Armstrong
Let Ln be the maximum number of regions we can get by drawing n (innite) lines in the
plane. We showed in class that
n
Ln = 1 + (1 + 2 + 3 + + n) = 1 +
k =1+
k=1
n2 + n + 2
n(n + 1)
=
.
2
2
1. Let f

Math 309
Homework 3 Solutions
Summer 2014
Drew Armstrong
Let k and n be integers such that 0 k n. Then we dene:
n
k
:=
n!
k! (n k)!
1. Use algebra to verify that for relevant values of k and n we have
n
k
=
n1
n1
+
.
k
k1
Proof. We nd a common denominator

Math 309
Homework 3
Summer 2014
Drew Armstrong
Let k and n be integers such that 0 k n. Then we dene:
n
n!
:=
k
k! (n k)!
1. Use algebra to verify that for relevant values of k and n we have
n
n1
n1
=
+
.
k
k
k1
2. Now give a counting argument for the ide

Math 309
Quiz 4 Solutions
Summer 2014
Drew Armstrong
1. Accurately state the Well-Ordering Axiom for N.
Every nonempty subset of N has a smallest element.
2. Accurately state the Principle of Induction for N.
Consider a function P : N cfw_T, F . If
P (b)

Math 309
Quiz 5 Solutions
Summer 2014
Drew Armstrong
1. Accurately state the the Principle of Induction for N.
Consider any function P : N cfw_T, F . If
P (b) = T for some b N, and
P (k) P (k + 1) for all k b,
then we have P (n) = T for all n b.
2. Cons

Math 309
Quiz 2 Solutions
Summer 2014
Drew Armstrong
Throughout this quiz, P and Q are Boolean variables.
1. Write out the truth table for P Q.
Proof.
P
T
T
F
F
Q P Q
T
T
F
T
T
T
F
F
P
T
T
F
F
Q P Q
T
T
F
F
T
F
F
F
2. Write out the truth table for P Q.
Pr

Math 309
Homework 4 Solutions
Summer 2014
Drew Armstrong
Recall that we dene the logical symbol by P Q := P Q. We pronounce
the statement P Q as if P then Q, or P implies Q. Recall that we say an integer
n Z is even if there exists k Z such tha n = 2k, an

Math 309
Homework 2
Summer 2014
Drew Armstrong
If S is a nite set then we let #S denote its number of elements. We call this the size or
the cardinality of S. Sometimes we will use the equivalent notation |S| := #S.
1. If S and T are nite sets, what is th

Math 309
Homework 4
Summer 2014
Drew Armstrong
Recall that we dene the logical symbol by P Q := P Q. We pronounce
the statement P Q as if P then Q, or P implies Q. Recall that we say an integer
n Z is even if there exists k Z such tha n = 2k, and we say t

Math 309
Homework 1
Summer 2014
Drew Armstrong
Let Ln be the maximum number of regions we can get by drawing n (innite) lines in the
plane. We showed in class that
n
Ln = 1 + (1 + 2 + 3 + + n) = 1 +
k =1+
k=1
n2 + n + 2
n(n + 1)
=
.
2
2
1. Let f and g be

Math 309
Quiz 3 Solutions
Summer 2014
Drew Armstrong
1. Accurately state the Binomial Theorem.
Proof. For any complex numbers x, y C and any natural number n N we have
n
(x + y)n =
k=0
n k nk
x y
.
k
2. Explicitly write out the expansion of (1 + x)6 . [Ev

Math 309
Homework 2 Solutions
Summer 2014
Drew Armstrong
If S is a nite set then we let #S denote its number of elements. We call this the size or
the cardinality of S. Sometimes we will use the equivalent notation |S| := #S.
1. If S and T are nite sets,