MAT 320W-03
04/18/14
160 points
Exam 3
Name
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1.(45) Let X = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Y = cfw_a, b, c, d, e, and let f : X Y be the function given
by
f = cfw_(1, c), (2, b), (3, b), (
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ive full credit
MAT 320W03
04/ 18/ 14
160 points
Exam 3
Show all work - unsupported answers may not race
1.(45) Let X : {1, 2,3, 4,5,6,7, 8,9, 10}, Y = {a,b, c, d, e}, and let f : X > Y be the function given
by
f = {(116), (27 b)1 (3: b): (
MAT 320W-03
03/14/14
140 points
Exam 2
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1.(15) Let an denote the number of dierent ways to divide a class on n students into two nonempty
groups, for n 2. One can show that an satises the
Weekly Homework 3
Molly Green
Foundations of Mathematics
January 26, 2016
Theorem 1.20. If a, b, and c Z, and a|b and b|c then a|c.
Proof. Let a, b, and c Z, such that a|b and b|c. Then b = ak and c = bl for some k, l Z
(Denition 1.10). Therefore,
c = bl
Weekly Homework 4
Molly Green
Foundations of Mathematics
August 30, 2014
Problem 1.48. Let A and B be two propositions. Conjecture an equivalent way of expressing the proposition (A B) as a proposition involving the conjunction symbol and
possiby the nega
Weekly Homework 5
Molly Green
Foundations of Mathematics
January 26, 2016
Denition 2.6. If A and B are sets, then we say that A is a subset of B, written A B,
provided that every element of A is also an element of B.
Theorem 2.11. (*, Transitivity of subs
Weekly Homework
Foundations of Mathematics
November 5, 2016
Theorem 2.36. Let S and T be sets. Then S T iff P(S) P(T ).
Proof. Two cases:
1) Proving from right to left.
Assume P(S) P(T ). Let x S. Then cfw_x S, so cfw_x P(S). Therefore, cfw_x P(T ),
so cf
Weekly Homework
Foundations of Mathematics
Theorem 3.6. (*) For all n N,
n
P
i=
i=1
n(n+1)
.
2
Proof. First, let n = 1. Then we get 1 = 1(1+1)
= 22 = 1 X
2
k
P
Next, assume for some k N, that
i = k(k+1)
is true.
2
i=1
Now, let n = k + 1. If
k+1
P
i=1
i=
(
Daily Homework
Foundations of Mathematics
November 5, 2016
Theorem 2.75. (*). Every closed interval is not an open set.
Proof. Let [a, b] be a closed interval, where a < b. We see that a [a, b]. If [a, b] were an
open set, then there would have to exist a
Daily Homework
Foundations of Mathematics
November 5, 2016
Theorem 2.100. (*). Let cfw_A be a collection of closed sets. Then
\
A
is a closed set.
T
Proof.
Let
cfw_A
be
a
collection
of
closed
sets.
Let
P
be
a
limit
point
of
A . Since
T
A for all . Beca
MAT 320W
2/3/2014 (due Friday, 2/7/2014)
25 points
HW #3
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1. Prove: If C A B, then A (A C) B.
2. Give an example to prove that the following statement is false.
If A C B C, then A B
MAT 320W
2/21/2014 (due Friday, 2/28/2014)
25 points
HW #4
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1. Let cfw_An | n N be a family of sets satisfying An An+1 for all n 1.
(a) Write a proof by mathematical induction that
MAT 320W-03
2/21/2014
Finite Topological Spaces
Denition Let X be a nite set. A topology on X is a subset T of the power set P (X ) of
X satisfying these three conditions:
(i) T and X T .
(ii) if U T and V T , then U V T .
(iii) if U T and V T , then U V
MAT 320W-03
02/14/14
125 points
Exam 1
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1.(10) Show that (x)(P (x) Q(x) is equivalent to (x)(P (x) = Q(x).
2.(20) True or false. If false, indicate a minor change (other than negation) th
MAT 320W
1/24/2014
Inference rules
Theorems and proofs in a formal system:
We x a universe U and a collection A of axioms, A = P1 . . . Pn . A theorem is then a proposition R such
that A R is a tautology. A proof of a proposition R is a sequence of propos
MAT 320W
1/17/2011 (due Friday, 1/24/2011)
25 points
HW #1
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1. Without constructing a truth table, determine whether the compound proposition is a valid
argument (i.e., is a tautolo
MAT 320W
1/27/2014 (due Friday, 1/31/2014)
25 points
HW #2
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1. Show that (x)(P (x) = Q(x) is equivalent to (x)(P (x) Q(x).
2. Denition Given integers m and n, we say m divides n if
Daily Homework
Foundations of Mathematics
Theorem 3.7. (*)
For all n N, 3 divides 4n 1.
Proof. Let n = 1. Then 4n 1 = 41 1 = 3. Since this is divisible by 3, we know that the
statement is true for n = 1.
Assume that the statement holds true for n = k. The