University of Toronto
Faculty of Arts and Sciences
Sample Examination, October 2012
MAT 246 H1F
Concpt Abstract Math
Instructor: Regina Rotman
Duration - 3 hours
No aids allowed
Total marks for this paper is 100
Please write your name in the space provide
(1) Prove that the set of functions f : R R has cardinality bigger than
R.
Solution
for a subset A R dene its characteristic function A by the formula
1 if x A
A (x) =
0 if x A
/
Its then clear that the map A A gives a 1-1 and onto correspondence between
Homework 2 with answers
Mat 246 Evening Section, Winter 2015
Due in tutorial on February 4 by 6:10 pm.
1. (Page 27 - 2a and 2b) For each of the following congruences, either nd a
solution or prove that no solution exists.
(a) 39x 13 (mod 5)
(b) 95x 13 (mo
Solutions to Practice Final 2
1. Using induction prove that
12 + 32 + . . . + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
3
Solution
First we verify the base of induction. When n = 0 LHS= 12 = 1 and RHS=
113
3
= 1.
Induction step. Assume the formula is true for n
Solutions to Practice Final 3
1. The Fibonacci sequence is the sequence of numbers F (1), F (2), . . . dened by the
following recurrence relations:
F (1) = 1, F (2) = 1, F (n) = F (n 1) + F (n 2) for all n > 2.
For example, the rst few Fibonacci numbers a
Practice Final 4
1. Use induction to prove that
1 + 2q + 3q 2 + . . . + nq n1 =
1 (n + 1)q n + nq n+1
(1 q)2
for any real q = 1 and any natural n.
2. (a) Find 43! (mod 45).
(b) Find the last digit of 32014 .
3. Let p1 , p2 be distinct prime numbers.
Prove
Solutions to Practice Final 1
1. (a) What is (20100 ) where is Eulers -function?
(b) Find an integer x such that 140x 133 (mod 301). Hint: gcd(140, 301) = 7.
Solution
(a) (20100 ) = (4100 5100 ) = (2200 5100 ) = (2200 2199 )(5100 599 ) =
= 2199 (2 1)599 (
Solutions to selected problems from homework 2
(1) Let p1 , p2 be distinct primes. Using the Fundamental Theorem of
Arithmetic prove that a natural number n is divisible by p1 p2 if and
only if n is divisible by p1 and n is divisible by p2 .
Solution
Its
Solutions to selected problems from homework 3
(1) Give a proof by induction of the following statement used class:
Let m 1 be a natural number. Then for any n 0 there exists an
integer r such that 0 r m and n r pmod mq.
Solution
We prove it by induction
1. Setting k = 1, we get 1 =
1(1+1)q+(1)q 2
(1q)2
Which we see is true as (1 q)2 = 1 2q + q 2 and q = 1
Assume the statement is true for k = n n n+1
i.e. 1 + 2q + 3q 2 + . + nq n1 = 1(n+1)q +nq
(1q)2
Then we have,
n
n+1
1 + 2q + 3q 2 + . + nq n1 + (n + 1)
(1) Using the Euclidean Algorithm prove that if gcdpa, bq 1 and a|c, b|c
then ab|c.
As gcdpa, bq 1 we know by euclidean Algorithm that there exists integers m, n such that am bn 1. Multiplying this equation by c we
get, acm bcn c. Now Since, a|c, we have
(1) Let p1 , p2 be distinct prime numbers.
Using the method from class give a careful proof of the formula
p pk1 pk2 q p pk1
1 2
1
pk 1qp pk pk 1q
1
2
2
1
2
2
Solution
Let n
The only prime divisors of n are p1 and p2 so if gcdpm, nq $
1 then either p1 |
Solutions to selected problems from homework 1
(1) The Fibonacci sequence is the sequence of numbers F (0), F (1), . . .
dened by the following recurrence relations:
F (0) = 1, F (1) = 1, F (n) = F (n 1) + F (n 2) for all n > 1.
For example, the rst few F
(1) Let a, b be odd integers.
Prove that a2 + b2 is irrational.
Hint: Look at divisibility by the powers of 2.
Solution
2 + b2 is rational. By a theorem prove in class.
Suppose a
n is
rational if and only if n is a complete square. Thus a2 + b2 = c2 for
s
Practice Final 2
1. Using induction prove that
12 + 32 + . . . + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
3
2. Let a, b, c be natural numbers.
(a) Show that the equation ax + by = c has a solution if and only if (a, b)|c.
(b) Find all integer solutions of 6x +
Practice Final 1
1. (a) What is (20100 ) where is Eulers -function?
(b) Find an integer x such that 140x 133 (mod 301).
Hint: gcd(140, 301) = 7.
2. (a) Prove, by mathematical induction, that 1 + 2 + 3 + . + n =
natural number n.
n(n+1)
2
for every
(b) Pro
(1) Let p1 , p2 be distinct prime numbers.
Using the method from class give a careful proof of the formula
p pk1 pk2 q p pk1
1 2
1
pk 1qp pk pk 1q
1
2
2
1
2
2
Solution
Let n
The only prime divisors of n are p1 and p2 so if gcdpm, nq $
1 then either p1 |
Solutions to Practice Final 3
1. The Fibonacci sequence is the sequence of numbers F (1), F (2), . . . dened by the
following recurrence relations:
F (1) = 1, F (2) = 1, F (n) = F (n 1) + F (n 2) for all n > 2.
For example, the rst few Fibonacci numbers a
Solutions to Practice Final 2
1. Using induction prove that
12 + 32 + . . . + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
3
Solution
First we verify the base of induction. When n = 0 LHS= 12 = 1 and RHS=
113
3
= 1.
Induction step. Assume the formula is true for n
Solutions to Practice Final 3
1. The Fibonacci sequence is the sequence of numbers F (1), F (2), . . . dened by the
following recurrence relations:
F (1) = 1, F (2) = 1, F (n) = F (n 1) + F (n 2) for all n > 2.
For example, the rst few Fibonacci numbers a
Solutions to the Term Test, Winter 2014
(1) (8 pts) Prove that there are innitely many prime numbers of the form 4k + 3.
Hint: If p1 , p2 , . . . pn are n such primes, look at 4(p1 p2 . . . pn ) 1.
Solution
Suppose there are only nitely many prime numbers
Solutions to Practice Final 1
1. (a) What is (20100 ) where is Eulers -function?
(b) Find an integer x such that 140x 133 (mod 301). Hint: gcd(140, 301) = 7.
Solution
(a) (20100 ) = (4100 5100 ) = (2200 5100 ) = (2200 2199 )(5100 599 ) =
= 2199 (2 1)599 (
MAT 246S
Solutions to the Practice Term Test 3
Winter, 2014
(1) Find a mistake in the following proof.
Claim: 1 + 2 + . . . + n = 1 (n + 1 )2 for any natural n.
2
2
We proceed by induction on n.
a) The claim is true for n = 1.
b) Suppose we have already p
MAT 246S
Solutions to Practice Term Test 1
Winter 2014
(1) Prove by mathematical induction that n3 + 5n is divisible by 6 for any natural n.
Solution
We rst check that the statement is true for n=1. We have 13 + 5 = 6 is divisible
by 6.
Suppose the statem
MAT 246S
Solutions to Practice Term Test 2
Winter 2014
(1) Find the formula for the sum 1 2 2 3 + 3 4 . . . + (2n) (2n 1) (2n) (2n + 1)
and prove it by mathematical induction.
Solution
Observe that (2n)(2n 1) (2n)(2n + 1) = (2n) (2) = 4n Thus we need to
n
Practice Final 1
1. (a) What is (20100 ) where is Eulers -function?
(b) Find an integer x such that 140x 133 (mod 301).
Hint: gcd(140, 301) = 7.
2. (a) Prove, by mathematical induction, that 1 + 2 + 3 + . + n =
natural number n.
n(n+1)
2
for every
(b) Pro
Practice Final 3
1. The Fibonacci sequence is the sequence of numbers F (1), F (2), . . . dened by the
following recurrence relations:
F (1) = 1, F (2) = 1, F (n) = F (n 1) + F (n 2) for all n > 2.
For example, the rst few Fibonacci numbers are 1, 1, 2, 3
Solutions to selected problems from homework 10
(1) (a) Prove that
3
is not constructible.
Solution
Suppose 3 is constructible. Then = 3 3 3 is constructible too. However, we know that is transcendental and
hence not constructible. This is a contradict
TUTOlOl
MAT 246, Quiz 1 Name
Part A: (2 marks) for two natural numbers n and m what does it mean m n?
f O C9
/cfw_A fxcrtghxt? XWmhr ( <5 WM b m a!) 2%)
Part B: (2 marks) Apply your exact defrw impartwAmteshow that 8l24
/,1 /\
4C9 (Bk ($4: 221 :3 3
I mi .1 t
MAT 246, Quiz 2 Name I; I7<~7WW ID TUTOlOl
Part A: (3 marks) complete the statements below: Principle of Mathematical lnduction': Assume S Is a
lw II
I; , I: N i M, 4:
subset of natural numbers with 1 E S and .V l01 i l
Ws
Well orderi
MAT246
.
Notes on Chapter 1
The purpose of these notes is to occasionally fill in some missing details from the proofs and arguments
presented in the textbook. Please note that the textbook is a very good source of intuitive reasoning yet
it misses many i
MAT246
Jan 20 Problem Session, Fri. 4:30-6, SS2102
Structure and techniques of proof
The first problem session we reviewed techniques of proof, and we saw a number of proof structures. This
Friday, Jan 20, in problem session 2, we will see more examples o
MAT246
Jan 13 Problem Session, Fri. 4:30-6, SS2102
Structure and techniques of proof
The first problem session reviews techniques of proof, and in general the culture of proof. Many relevant
terminologies and logcal arguments for making inferences will be