Connor Galvin
11/29/2016
Congruence Proof
If x0 is a solution to ax b mod n and y x0 mod n, then y is also a solution to ax b mod n. Is the
converse true?
Proof: Let x0 be a solution to ax b mod n. Al
October10,2014
1) LetA,B,andCbesetsandletS=A\(B C)andT=(A\B) (A C). Proveor
disprove:
a) ST
Proof:LetxS.ThismeansthatA\(BC)andthatxAbutx(BC).Sincewe
havethatx(BC),thenxBandxC.SayxisanelemetofAbutxB;we
October 20, 2014
Prove that for any indexed collection of sets,
We want to prove that
First, Let x
We must prove that
. Then, x
Ac for any Ac ( L). Thus, x
and
. Therefore, there exists A ( L), x
ProofHW
October24,2012
i.
Letf:NxZ Rbegivenbyf(x,y)=xy
a) Whatisf(2,2)?Whatisf(1,2)?f(2,1)?
f(2,2)=22=4
f(1,2)=12=1
f(2,1)=21=
b) Whatisf1[cfw_1]?
cfw_x=1andy=allintegers0
c) Whatistherangeoff?
(,)
ii
Math 239: Intro to Mathematical Proof
Due: Wednesday, September 3, 2014
SUBMIT your typed proof to one and only one (your choice) of the following. Use a
verbal proof like that on page 8 for Theorem 1
GROUP A
September 12, 2014
If x is odd, then 8 | (4x2 + 12)
Proof: Let x be an odd integer say 2n+1 where n is some integer. So, 4x2 + 12 = 4 (2n + 1)2 + 12
= 4 (4n2 + 4n + 1) + 12 = 16n2 + 16n + 4 +1
November 6, 2014
Let f: S T and let A and B be subsets of S. Prove or disprove
i.
If f is surjective, then f (A B) = f (A) f (B).
Counterexample:
S
T
O
O
O
O
O
O
O
ii. If f is injective, then f (A B)
GROUP A
Problem ic: ac bd mod n
Proof: Suppose a b mod n and c d mod n. Let q be such that a b = qn for some integer q
and let p be such that c d = pn for some integer p. Then, a = qn + b and c = pn +
mine In: injwivity and the surjttlivily 0f [ht fallnwing functinns. Indi
;.r. at. e n Jgt-tttiieiqn-+ 2
f: NI- w dened byn) = Lna'EJ
f:R1-Rdenedhyx} =.rt2 - xjferxnndx] = .rfeer
ine a funetien from N
Math 239 Introduction to Proof
Course Information Course: 18019 Math 239 Section 05
Time and Place: MWF 12:30-1:20 Sturges 105
Instructor: Olympia Nicodemi
Contact Information:
Oce: South Hall 325B
Intro to Proofs Test 1 Study Sheet
P
Q
P Q
P Q
PQ
PQ
T
T
T
T
T
T
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
F
T
T
Tautology- all true
Contradiction- all false
Natural numbers (N): 1, 2, 3, 4
Integers (Z): -2, -1,
Quiz 1
Name .
1. Is P Q equivalent to (P Q) ? Justify your answer.
2. Is P Q equivalent to P Q? Justify your answer.
3. Prove verbally: If P (Q R) is true then (P Q) (P R) is true.
4. Write the negati
October 2, 2014
Determine which is true (if any) and prove your results. Note: an answer of false
requires a counterexample
a) For any sets S, T, and W, S \ (T \ W) = (S \ T) \ W
False, Let S = cfw_,
Spindle Problem
Proof:
The formula for the minimum number of moves with three pegs and n spindles is 2 n 1. The
recurrence relation that leads to this is,
tn = 2tn 1 + 1
Where tn is the number of move
Connor Galvin
12/6/2016
Cardinalities Part 2
Problem 7:
Proposition: Suppose that A and B are disjoint sets, both of cardinality 0. Prove |A B| = 0.
Play: In order to get |A B| = 0, we must find a way
Connor Galvin
12/4/2016
Cardinalities
6.
Proposition: Suppose that A and B are disjoint sets such that |A| = 0 and |B| = n. Prove that |A B|
= 0.
Proof: Let |A| = 0, and |B| = n, such that both sets a
Connor Galvin
Intro to Proofs
Nicodemi
Verbal Proof
Demorgans Laws
~(p and q) <=> ~p or ~q
Assume that ~(P and Q) is true, so (P and Q) is false. Since (P and Q) is false, either P is false or Q
is fa
Connor Galvin
NICO pg 176
41.
a) cfw_x: x is a natural number divisible by 2 or 3
Bijection: a bijection exists if
f(a)=a when a is even
f(a)=a/3 when a is odd
b) cfw_x: x is real and x - LxJ = 0 or 0
Connor Galvin
NICO HW pg 217
Question 8:
1) Some related pairs include:
(3,9) R (1,3), (4,8) R (1,2), (1,2) R (4,8), (1,1) R (1,1)
2) Prove that this is an equivalence Relation:
Proof:
First we must s
Connor Galvin
Roomy sets hw
Nicodemi
3. Proposition: If A and B are roomy sets, then A B is roomy.
c
Proof: Let A and B be sets, such that both are roomy sets. If A is a roomy set, then for every x A,
Connor Galvin
Professor Nicodemi
8/30/2016
Assignment #1
1,2,3,4,6b
1.
a)
b)
c)
d)
e)
f)
g)
Prop
Prop
Not prop
Not prop
Prop
Prop
Not prop
a)
b)
c)
d)
e)
f)
Mary is not tall
The Cat is black
Some dogs
Extra Credit
iii. Prove that n curves separate a paper into (n2 + n + 2)/2 regions, provided that any two curves
meet exactly once on the paper, and no three curves meet at exactly one point.
Proof: F
November 7, 2014
Test Correction:
5. Suppose that b and y are natural numbers. Let A = cfw_x Z: x y mod b and let B =
cfw_x Z: x2 y2 mod b. Prove or disprove the following.
b) B A. Counter example:
Le
November 7, 2014
31. Show that a function f: A B is surjective if and only if f (A) = B.
Proof: In case one, we have to show that both f: A B is surjective and that f (A) =
B. A function is considered
Exam 2 Review
1. Let A = cfw_a, cfw_a, b. Answer true or false.
(a) A has two elements.
(b) cfw_a P (A)
(c) cfw_a P (A)
(d) cfw_a P (A)
(e) cfw_a P (A)
(f) cfw_, b P (A).
2. Prove via a verbal (elemen