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 p
(1) Let S = (0, 1) and T = [0, 1). Let f : S T be given by f (x) = x
and g : T S be given by g(x) = x+1 .
2
(a) Find SS , ST , S , TS , TT , T
(b) give an explicit formula for a 1-1 and onto map h : S
(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 t
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 tha
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.
Fo
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
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 (2
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 ) = (22
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. W
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) Th
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 transcend
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 o
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 d
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 ) = (22
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
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
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
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 o
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
(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
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.
So
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
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
Homework 6 Answers
Mat 246 Evening Section, Winter 2015
Not to be collected due to the strike
1. (Page 153 - 1 (a), (j), (r) Determine which of the following numbers are
constructible.
(a)
1
3+ 2
Ans
Department of Mathematics
University of Toronto
MAT246H1S L5101 Midterm Examination
Concepts in Abstract Mathematics
Examiner:
Peter Rosenthal
LAST NAME:
FIRST NAME:
STUDENT NUMBER:
.
.
.
.
.
There ar
Department of Mathematics
University of Toronto
MAT246H1F L5101 Final Examination
Concepts in Abstract Mathematics
Examiner: Peter Rosenthal
Monday, December 9, 2013
LAST NAME:
FIRST NAME:
STUDENT NUM
Notes on Cardinality
Mat 246 Evening Section, Winter 2015
Denition 1. The sets S and T have the same cardinality if there is a function
f : S ! T that is one-to-one and onto all of T .
Notation 2. We
(1) Let p1 , p2 , p3 be distinct prime numbers.
Using the method from class give a careful proof of the formula
ppk11 pk22 pk33 q ppk11 pk11 1 qppk22 pk22 1 qppk33 pk33 1 q
(2) Let a, b, c be natural
MAT 246S
Practice Term Test 3
Winter 2014
(1) Find a mistake in the following proof.
Claim: 1 + 2 + . . . + n = 12 (n + 12 )2 for any natural n.
We proceed by induction on n.
a) The claim is true for
University of Toronto
MAT246 midterm test
Friday Feb. 26, 2016
Duration: 100 minutes
No aids allowed
Instructions: This exam contains 105 marks, out of which 5 marks are bonus
marks. Please answer all
MAT246 Midterm Review
Part 1 Induction, Well Ordering Principle and natural numbers
Definitions:
1) Divisibility: a,b are natural numbers (/integers) a|b if there is a natural number(/integer) k
such
(1) Let P (z) be a polynomial with complex coefficients such that P (n) =
0 for all integer n.
Prove that P (z) = 0 for all z C.
(2) Express the following complex number as a + bi for some real a, b
(