Foundations of Mathematics (MATH 6100) Midterm
Problems
1. (4 points) Populate the following truth table.
P
T
T
F
F
Q
T
F
T
F
P Q
P
P
T
T
F
F
Q
T
F
T
F
P Q P
T
F
F
F
T
T
T
T
Q
P
Q
T
F
T
T
P
T
T
F
F
F
T
F
F
Q
F
T
F
T
Solution.
Q
T
F
T
F
2. (10 points) Una
Foundations of Mathematics (MATH 6100) Homework #7
Problems
1. If A, B, C are sets, prove that A (B C) = (A B) (A C).
Proof.
(x, y) A (B C) x A y B C
x A (y B y C)
(x A y B) (x A y C)
(x, y) A B (x, y) A C
(x, y) (A B) (A C).
2. If cfw_Ai : i I and cf
Foundations of Mathematics (MATH 6100) Homework #2
Remarks
By a counterexample, I mean a specic example that satises the
premises but denies the conclusion. For example, to disprove A B
(which is equivalent to the implication x A x B), it suces
to exhibi
Foundations of Mathematics (MATH 6100) Homework #1
Remarks The precedence of logical connectives is: (1) negation, (2) conjunction, (3) disjunction, (4) conditional, (5) biconditional. Please order the
truth values of the premises of your truth tables in
Foundations of Mathematics (MATH 6100) Homework #3
Remarks In problems 2-4, extra points will be awarded if you convince
me of your answer. In problems 9-11, F and G denote families of sets. To
analyze the logical form of a statement means to rewrite it i
Foundations of Mathematics (MATH 6100) Homework #4
Problems
1. Analyze the logical form of x P(A B).
2. Analyze the logical form of
F (A ( G).
3. (T/F): A B if and only if P(A) P(B).
1 1
4. Let the universe be R. For each n N, dene An := ( n , n ). Use
De
Foundations of Mathematics (MATH 6100) Homework #5
Problems
1. Prove that 2n n + 1 for all n N.
2. Prove that 2n > n for every n N.
3. Prove that for all n N,
n
i2 =
i=1
n(n + 1)(2n + 1)
.
6
4. Let r be a real number, r = 1. Prove that for every n 1,
1 +
Foundations of Mathematics (MATH 6100) Homework #6
Problems
1. Prove that for all n N,
n
(i 1)i =
i=1
n(n 1)(n + 1)
.
3
2. Suppose that cfw_An is a sequence of sets such that An An+1 for every
n=1
n. Use induction to prove that for all n N,
n
Ai = An .
i
Michael J. Fairchild
1
Cardinality
How do we measure the size of a set? For nite sets, one can just count
the number of elements, but what about innite sets? The notion of size,
or cardinality, of a set can be dened in terms of injections, surjections, an
Foundations of Mathematics (MATH 6100) Homework #10
Problems
1. (a) If g f is surjective, which if any of f or g must be
surjective? Justify your answer.
(b) If g f is injective, which if any of f or g must be
injective? Justify your answer.
Solution. Thr
Foundations of Mathematics (MATH 6100) Homework #8
Remarks The following denitions will be used in problems 6 and 7.
Let be a partial order relation on a set X, and let A X. An element
y X is an upper bound for A if x y for all x A. It is the least upper
Foundations of Mathematics (MATH 6100) Homework #2
Remarks
By a counterexample, I mean a specic example that satises the
premises but denies the conclusion. For example, to disprove A B
(which is equivalent to the implication x A x B), it suces
to exhibi
Foundations of Mathematics (MATH 6100) Homework #6
Problems
1. Prove that for all n N,
n
(i 1)i =
i=1
n(n 1)(n + 1)
.
3
Proof. By induction. Consider the base step (n = 1). On the one hand,
1(11)(1+1)
1
= 102 = 0. So
i=1 (i 1)i = 0 1 = 0. On the other han
Foundations of Mathematics (MATH 6100) Homework #9
Problems
1. Suppose f : R R is injective. Is g : R R : x ef (x) necessarily
injective? If so, prove it. If not, exhibit a counterexample.
Proof. Yes. It was proved in class that the function x ex from R i
Foundations of Mathematics (MATH 6100) Homework #4
Problems
1. Analyze the logical form of x P(A B).
Solution.
x P(A B) x A B
y(y x y A B)
y(y x y A y B)
2. Analyze the logical form of
Solution.
F (A (
F (A ( G).
G) x(x
F xA(
G)
x(A F(x A) x A x
G)
Foundations of Mathematics (MATH 6100) Homework #3
Remarks In problems 2-4, extra points will be awarded if you convince
me of your answer. In problems 9-11, F and G denote families of sets. To
analyze the logical form of a statement means to rewrite it i
Foundations of Mathematics (MATH 6100) Homework #5
Problems
1. Prove that 2n n + 1 for all n N.
Proof. By induction. The base step (n = 1) is clear. For the induction step,
assume 2n n + 1. Then 2(n + 1) = 2n + 2 (n + 1) + 2 > (n + 1) + 1.
So, the inequal
Foundations of Mathematics (MATH 6100) Final Exam
Name:
Problems
1. (2,2,1 points) Determine if the statements P Q and P Q are equivalent by constructing truth tables for each.
Solution.
P
T
T
F
F
Q
T
F
T
F
P Q P
T
F
F
F
T
T
T
T
T
F
T
T
Q
T
F
T
F
From the
Foundations of Mathematics (MATH 6100) Homework #1
Remarks The precedence of logical connectives is: (1) negation, (2) conjunction, (3) disjunction, (4) conditional, (5) biconditional. Please order the
truth values of the premises of your truth tables in
Michael J. Fairchild
1
Functions
One usually thinks of a function (or map) as a rule that assigns to
each element in one set a unique element in another set. Functions
are absolutely ubiquitous: the temperature depends on the time of
day, your age depends