Concepts of Mathematics
Homework 1 Solutions
Problem (1.7). The statement is not always true for x, y R. Give an example
where it is false, and add a hypothesis on y that makes it a true statement.
If x and y are nonzero real numbers and x > y , then (1/x
Concepts of Mathematics
Homework 2 Solutions
Problem (2.3). Consider the following sentence: If a is a real number, then
ax = 0 implies x = 0. Write this sentence using quantiers, letting P (a, x)
be the assertion ax = 0 and Q(x) be the assertion x = 0. S
Concepts of Mathematics
Homework 7 Solutions
Problem (6.45, graded). The royal treasury has 500 7-ounce weights, 500 11ounce weights, and a balance scale. An envoy arrives with a bar of gold, claiming
it weighs 500 ounces. Can the treasury determine wheth
Concepts of Mathematics
Homework 6 Solutions
Problem (6.1). Explain why the following makes no sense: Let n be relatively
prime.
Proof. Relatively prime is a property of a pair of numbers, not a property of
a single numberas prime is.
Problem (6.8, graded
Concepts of Mathematics
Homework 4 Solutions
Problem (4.9). Decide whether the following statement is true or false; justify.
If f and g are monotone functions from R to R, then g f is also monotone.
Proof. TRUE. WLOG, both f and g are nondecreasing. Cons
Concepts of Mathematics
Homework 3 Solutions
Problem (3.5, graded). True or false: For n N,
n
k=1 (2k
+ 1) = n2 + 2n.
Proof. TRUE. Proof by induction:
1
Base Case: For n = 1, we have k=1 (2k + 1) = 2(1) + 1 = 3 = (1)2 + 2(1).
Hence the formula holds for n
MATH 127: Exam 3b Solutions
Wednesday, April 20, 2011
1. (a) Observe that primes which appear in the prime factorization of n must have power at
least 2, otherwise wed have a prime dividing n with its square root not. Some of these
powers are even, and so
Full Name:
MATH 127: Exam 3B
Section:
Wednesday, April 20, 2011
You are expected to justify your answers in a manner that an average 21-127 student could
understand. This may require sentences for clarication.
Your work should be neat and organized and