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: First we show that P(1) is true, namely
P(1) = (12 + 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 +16 = 16n2 + 16n + 16 = 16 (n2 + n + 1). Therefore, 8 |
(
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) = f (A) f (B).
Proof: Let y f (A B). There exists some
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 + d. Using
algebra, we obtain:
ac = (qn + b) (pn + d)
=
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 to N that is:
lnjeetive hut net smjeetive
Surjeetive bu
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
e-mail Nicodemi@geneseo.edu
Oce Phone: X5390
Math Dep
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, 0, 1, 2
Rational Numbers (Q)
Real Numbers (R)
If all p
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 negation, the converse, and the contrapositive of the followi
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.
Suppose that P (Q R) is true. Prove that (P Q) (P R)
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. Letf:S T.LetAbesubsetofS,andCandDbesubsetsofT.Proveor
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:
Let b = 4, x = 3 and y = 5
x y mod b b | (x y)
x2 y2 mod
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 surjective when all the elements in B can find at leas
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 moves with n spindles present.
First we show that t1 is tru
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_, -20, -19, -18, -17, -16
Let T = cfw_, -18, -15, -12, -
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;wecansay
thatx(A\B).Also,saythatxAbutxC,(AC)holdsbecausex
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 A. But then x
. Second, Let x
( L). Thus x A for any A
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 (element chasing) proof or disprove with a counter example.
(a