Brianna Camp
Conjecture 7A. If S and T are the sets defined by S = cfw_x Z | x 5 (mod 7) and T = cfw_x
Z | x2 4 (mod 7), then T is a proper subset of S.
The integer x = 2 is a counterexample that proves this conjecture is false. In order for T S,
the fol
Brianna Camp
Theorem 8B. For any sets X and Y in a universal set U , (X Y c ) (Y X c ) = X if and only
if Y = .
Proof. First we will prove that if (X Y c ) (Y X c ) = X, then Y = by way of contradiction.
In particular, we assume (X Y c ) (Y X c ) = X and
Brianna Camp
Theorem 9A. The function g : R cfw_3 R cfw_2 defined by g(x) =
onto.
Proof. Let g : R cfw_3 R cfw_2 be defined by g(x) =
2x+1
x3
is both one-to-one and
2x+1
x3 .
We will first prove that g is one-to-one. By definition, this means we must show
Brianna Camp
Theorem 10E. For each n Z, n is the sum of 8 consecutive integers if and only if n 4
(mod 8).
Proof. First, we will prove that if n is the sum of 8 consecutive integers, then n 4 (mod 8). To
this end, we assume that n is the sum of 8 consecut
Brianna Camp
Theorem 6B. For all natural numbers n,
n
X
i=1
4i2
n
1
=
.
1
2n + 1
Proof. We will proceed using Mathematical Induction.
Base Case: When n=1, notice that the left side of the equation is
1
2(1)+1
1
4(1)2 1
=
1
3
and the right side of
1
3.
=
T
Brianna Camp
Theorem 5A. For each natural number n, the nth derivative of f (x) = x2 ex is
f (n) (x) = [x2 + 2nx + n(n 1)]ex .
Proof. We will proceed using Mathematical Induction.
Base Case: When n=1, the statement P (1) is
f (1) = [x2 + 2(1)x + 1(1 1)]ex
Brianna Camp
Theorem 4B. The real number 14 42 is irrational.
Proof. To prove
that
14 42 is irrational, we will use a proof by contradiction. In particular,
we assume that 14 42 is rational. This tells us that there exist integers a and b with b 6= 0
su
Brianna Camp
Conjecture 3B. Let k be an integer. If k 2 + 1 is even, then k 2 1 is divisible by 4.
Theorem 3B. If k 2 + 1 is an even integer, then k 2 1 is divisible by 4.
Proof. We will show that if k 2 + 1 is an even integer, then k 2 1 is divisible by
Brianna Camp
Conjecture 2A. If k is a natural number and k 2, then (2k, k 2 1, k 2 + 1) is a triangular triple.
This proposition is false as is shown by the following counterexample: Letting k = 2, observe
that it follows that
(2k, k 2 1, k 2 + 1) = (2(2)
Brianna Camp
Conjecture 1A. If a is a Type 2 integer and b is a Type 1 integer, then a3 + 2b2 is a Type 2
integer.
This proposition is false as is shown by the following counterexample: By the definition of Type
1 and Type 2 integers, there exist integers