Download Objectives - Stony Brook Mathematics

Weekly Objectives for Math 200
December 5, 2016
1
Week 1
• Write truth tables and Venn diagrams for logical expressions.
– Show using truth tables that (P ∧ Q) =⇒ (P ∨ Q)
– Write a truth table for (¬p∨q)∧(q =⇒ ¬r ∧¬p)∧(p∨r). Is there any
combination of truth values for p, q, r making the statement true?
• Translate English language statements into symbolic expressions.
– Formalize the sentence “Aldo is Italian or if Aldo isnt Italian then
Bob is English.”
– Formalize the sentence “Exactly two, among Aldo, Bruno and Carlo
passed the exam.”
• Solve little logic puzzles and explain your solutions.
• Distinguish between universal and existential statements.
• Form the negation, converse, and contrapositive of statements.
– Formalize the sentence “Carlo comes to the party provided that Davide doesnt come, but, if David comes, then Bruno doesnt come” and
write its negation both in words and symbols.
– What is the contrapositive of (P ∨ Q) =⇒ (R ∧ S)?
2
Week 2
• Direct proof of basic properties of (ordered) fields starting from axioms
(you will be given a cheat sheet of axioms).
– Property of zero: xy = 0 ⇐⇒ (x = 0 ∨ y = 0)
– The multiplicative identity is unique: (∀a, ax = a) =⇒ x = 1
– Prove commutativity of addition (a + b = b + a) without using the
commutativity axiom AG1. (i.e. the field axioms in the book are not
independent!)
1
– If x > 1 then x−1 < 1.
– Uniqueness of multiplicative inverses: if ax = 1 and ay = 1 then
x = y.
• Look at addition or multiplication tables and verify the identity (zero),
commutativity, associativity and subtraction/division axioms.
– Consider the three multiplication tables
·
1
s
s2
t
ts ts2
2
1
1
s
s
t
ts ts2
2
2
s
s
s
1 ts
t
ts
1
s
ts ts2
t
s2 s2
t
t
ts ts2 1
s
s2
t
s2
1
s
ts ts ts2
2
2
ts ts
t
ts
s
s2
1
· 1 s
1 s 1
s 1 s
· 1 s t
1 1 s t
s s 1 1
t t 1 1
Each of these fails one of the identity, commutativity, associativity,
or division axioms. Which is which and why?
• Prove or disprove statements about divisibility of integers, possibly by
reduction to cases.
– Show that 3 divides x3 + 6x2 + 5x for all x ∈ Z
3
Week 3
• Prove simple results about the nonexistence of solutions to integer equations.
√
Prove that 3 is irrational. Prove that the product of a rational and
an irrational number is irrational.
• Other very simple proofs by contradiction.
– Prove there is no smallest positive real number.
– Consider the set of all pairs of real numbers with component-wise
addition and multiplication:
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac, bd)
where the identity 1 = (1, 1) and 0 = (0, 0). Show that this does not
satisfy the field (division) axiom.
2
4
Week 4
• Use induction to prove facts about sums and sequences.
– Prove that 3 divides 7n − 10 for all n ∈ N
– Find a simple expression for the sum:
1+
1 1
1
+ + ··· + n
2 4
2
in terms of n and prove that it works using induction.
– Prove
12 − 22 + 32 − 42 + · · · + (−1)n+1 n2 =
1
(−1)n+1 n(n + 1)
2
• Understand inductive definitions, such as of the Fibonacci and Catalan
numbers.
– Define polynomials Tn (x) inductively by:
T0 (x) = 1
T1 (x) = x
Tn+1 (x) = 2xTn (x) − Tn−1 (x)
What is T4 ?
5
Week 5
• Understand simple statements with multiple quantifiers, and be able to
prove disprove and negate them as appropriate.
– Is the given expression true or false? Explain your answer.
∀x ∈ R, ∃y ∈ R|x + y = 0
∃x ∈ R|∀y ∈ R, xy < 1
∃x ∈ N|∀y ∈ N, y < x
∃x ∈ R|∀y ∈ R, 2x − x2 > 2y − y 2
∃x ∈ R|∀y ∈ R, 2x − x2 ≥ 2y − y 2
– Negate the statement:
∀ > 0 ∈ R, ∃δ > 0 ∈ R|∀x ∈ R, (x + δ)2 − x2 < Which of the two statements (the original or its negation) is true?
3
6
Week 6
• Be able to read and understand set theory notation.
– Given A ( C and A ⊆ B ⊆ C prove (A ( B) ∨ (B ( C).
– Prove that (A − B) ∩ (B − A) = ∅ for any two sets A, B.
– Let A, B ⊆ R. Prove (A ∪ B) ∩ (A ∩ B)c = (A − B) ∪ (B − A).
– (Stolen from a Math Overflow post?) Suppose A ∪ B = C, (A ∪ C) ∩
B = C, and (A ∩ C) ∪ B = A. Show A = B = C.
– Write statements equivalent to ∀x ∈ R, f (x) = 0 and ∃x ∈ R|f (x) =
0 using set notation but without using quantifiers.
• Be aware of the existence and importance of axiomatic set theory.
– What is Russell’s Paradox and how do we avoid it (short answer, two
or three sentences only)?
– What is the name of the commonly accepted set theory axioms?
• Cartesian products and power sets.
– Let A ⊃ B ⊃ C be sets with |2A−B ∪ 2B−C ∪ 2C | = 22. Find
|A|, |B|, |C|.
7
Week 7
• Basic vocabulary: domain, codomain, image; injective, bijective, surjective.
• Understand the definition of function.
• Injective, surjective, and bijective.
• Composition of functions.
– Prove or disprove: if f, g : R → R are polynomial functions then
f ◦ g = g ◦ f (i.e. composition of polynomials is commutative).
– Let p : R → R be a polynomial function that is bijective with degree
> 1. Show that the inverse function p−1 is not a polynomial.
– Let f : A → B, g : B → C be two bijective functions show that
(g ◦ f )−1 = f −1 ◦ g −1 .
– Recall that for a bijective function f f −1 is defined by the property
that f ◦ f −1 = Id and f −1 ◦ f = Id, where Id is the identity function
x 7→ x. Find two non-bijective functions f, g such that f ◦ g = Id
(but g ◦ f 6= Id).
– Prove that the composition of two surjective functions is surjective.
4
• Graphs of functions.
– Let A = R, B = R, G = {(x, y) ∈ A × B|x = y 3 − y}. Is G the graph
of a function A → B? Is G the graph of a function B → A?
8
Week 8
• Basic counting: sums and products
– Compute |{n ∈ N10000 |(3 divides n) ∧ ¬(5 divides n)}|
• Inclusion-Exclusion
– How many surjective functions are are there from N5 to N3 ?
– (Stolen from notes by Frdrique Oggier) In a fruit feast among 200
students, 88 chose to eat durians, 73 ate mangoes, and 46 ate litchis.
34 of them had eaten both durians and mangoes, 16 had eaten durians
and litchis, and 12 had eaten mangoes and litchis, while 5 had eaten
all 3 fruits. Determine, how many of the 200 students ate none of
the 3 fruits, and how many ate only mangoes?
• Pigeonhole principle
– How many 11 digit numbers are there with no repeated digit?
– Given five points inside an equilateral triangle of side length 2, show
that there are two points at most 1 unit apart.
– Let p : Nn → Nn be a permutation. Write pn for the composition of
p n times (i.e. p1 = p, pn+1 = p ◦ pn ). Show that there are natural
numbers l, m such that pm = pn . Moreover show there is a natural
number n such pn is the identity function (i.e. pn (i) = i for all i).
– Show that among any four numbers {a1 , a2 , a3 , a4 } there are at least
two (ai , aj ) with the difference ai −aj a multiple of 3. (Hint: consider
the remainder after dividing each ai by 3.)
9
Week 9
• Unique behaviour for infinite sets.
– Let f : X → X be an injection. Show that |X − Im f | = | Im f −
Im f ◦ f |.
– Show |Z| = |N|.
– Is the function {x ∈ R|x ≥ 0} → [0, 1) with rule x 7→
Show |(−1, 1)| = |R|.
• Prove that |X| < |2X |
5
x
x+1
bijective?
10
Week 11: Binomial coefficients
• What is the coefficient of x5 in (2x − 3)7 ?
• How many subsets of N11 have a prime number of elements?
• How many ways are there to arrange 3 identical red balls, 4 identical blue
balls, and 5 identical yellow balls in a row?
Pn
• Prove k=0 kc = n+1
c+1 for all natural numbers n and c.
• (Assume common facts about prime numbers, in particular that a prime
divides a product only if it divides one of the factors.) Show that if p is
prime then p divides kp for all k with 0 < k < p.
11
Week 12: Equivalence Relations
• Does the subset
S = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)}
of N4 × N4 define an equivalence relation (by x ∼ y ⇐⇒ (x, y) ∈ S)? If so
what are the equivalence classes?
• Show x ∼ y ⇐⇒ x4 = y 4 is an equivalence relation on Z13 . What are
the equivalence classes? If [a] is an element of Z13 denote the equivalence
class of [a] in Z13 / ∼ by [a]∼ . Is the operation [a]∼ + [b]∼ = [a + b]∼ well
defined? How about [a]∼ [b]∼ = [ab]∼ ?
• How many equivalence relations are there on N10 with exactly 2 equivalence classes?
• Is
(a, b) ∼ (c, d) ⇐⇒ a2 d3 = c2 b3
an equivalence relation on (R − {0}) × (R − {0})?
12
Week 13: Modular arithmetic.
• Recall that Z7 is a field (since 7 is prime). Find [10]−1 in Z7 .
• Let x = an an−1 . . . a0 be a number written in decimal notation. Show
that x is divisible by 7 if and only if
an . . . a1 − 2a0
is. For example, we can check 581 is divisible by 7 since 58 − 2 · 1 is. (Hint:
show
[an . . . a0 ]7 = 3[an . . . a1 − 2a0 ]7 .
)
6
• Let n be any natural number greater than 1. Show that for any [x] ∈ Zn
there are a, b ∈ Nn+1 such that [x]a = [x]b . Show that if Zn is a field then
there is some c ∈ Nn , c > 1 such that [x]c = [x].
• Let n ∈ N, n 6= 0 be any nonzero natural number. Show that there is some
nonzero multiple of n that can be written using only the digits 0 and 1.
(Hint: pigeonhole with the numbers 1, 11, 111, . . . .)
7