2 For each of the following statements decide whether it is TRUE or FALSE. If it is TRUE, then
prove it. If it is FALSE, then provide a counter-example.
Let f and g be functions from R to R.
(a) If f and f g are bounded, then f + g i
1.7. The domain of the absolute value function is all of R since R = (-, 0] [0, ). The image I is S = cfw_x R : x 0. If x S, then |x| = x, so x I. If x S, then x < 0, so |y| = x for any y R since the definition of the absolute value / function ensures tha
4.12 (a) False. One counterexample is f (x) = e-x . (b) False. Take f (x) = 0. (Any constant function will work.) (c) False. Consider x if 0 = x = 1, 1 if x = 0, f (x) = 0 if x = 1. (d) True. If |f (x)| M for every x R, then M + 1 is not in the image of f
MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2010 Solutions to Selected Problems from Problem Set D
For the first two questions (about fields) see solutions to Quiz #2. 2.2. Take a = 0 and b = 1. The statement claims that there are integer
MAT102S - Introduction to Mathematical Proofs - UTM Problem Set A - Spring 2010
Here are the Exercises assigned: 1.2. Fill in the blanks. The equation x2 +bx+c = 0 has exactly one solution when and it has no solutions when . 1.5. (-) Consider the Celsius
1.13. Let x A. Then x = 2k - 1 for some k Z. Let k1 = k - 1. We then have 2k1 + 1 B, but also 2k1 + 1 = 2(k - 1) + 1 = 2k - 1 = x, so x B. Hence A B. Conversely, let x B so that x = 2m + 1 for some m Z and let m1 = 2m - 1. Then 2m1 - 1 A and x = 2m + 1 =
MAT102:Mathematical and logical-Lecture 16: Ch. 4 Bijections and Cardinality
Injections surjections and bijections
Example: consider the following function F:AB, given by
Question: if we reverse arrows, do we get a function from B to
MAT102: Mathematical and logical-Lecture 13: Chapter 3 Induction (continue)
Question: How many subsets do a set S with n elements have (including S and )?
This is Not a statement (cant be proven).
For n=2, S=cfw_a,b, then the subsets are cfw_a, cf
Mat102:Mathematical Thinking-lecture 6:function and field
Definition: a function from a set A to a set B assigns to each aA , a unique element
f(x) B(called the image of a function f).
A is called thd domain of f.
B is called the target(also range) of f.
Mat102:Mathematical Thinking-Lecture 8: Ch. 2 Language and Proofs
In order to communicate and argue in mathematics properly, we need to use the
language of mathematics (i.e its syntax) correctly.
Understand various components of the language and put them
MAT102:Mathematical and logical-Lecture 20: Modular Arithmetic Relations
Often, we need to compare two objects in a given set, and decide whether they do or
do not satisfy a certain property ( or condition).
Do they go to the same university, gi
MAT102: Mathematical and logical-Lecture 14: Chapter 3 Induction (continue)
- Prove that for any a checkerboard 2nnN
2 ,n with one square removed has an Ltiling (i.e can be covered by L-shapes
Proof : Base case: for n=1, we have 2
Mat102:Mathematical Thinking-Lecture 9: Truth and Falsity
From last time:
e.x Lets translate a math statement into symbols. Every integer is either even or odd.
Let E be the set of even integers and O be the set of odd integers.
1. a translate will be ( x
MAT102: Mathematical and logical-Lecture 15: Chapter 3 Induction (continue)
- Prove that for any
checkerboard with one square removed has an
L-tiling (i.e can be covered by L-shapes ).
Proof : Base case: for n=1, we have 22 board with 1 square
MAT102: Mathematical Thinking-Lecture 5: Sets
The Internal Notation:
If a, b R with a b, we define:
[ a , b ] = cfw_ x Ra x b (closed interval)
( a , b ) =cfw_ x Ra< x< b (open interval)
Set operations: For sets A and B, we define:
Intersection: A B=cf
MAT102:Mathematical and logical-Lecture 21: Modular Arithmetic Relations
Define the following equivalence relation on Z: ab iff a-b is divisible by 4.
MAT102:Mathematical and logical-Lecture 17: injection,surjection and bijection
Definition:let f:AB and g:BC be two functions. The composition of g with f,
denoted g o f, is the function from A to C, given by (g o f)(x)=g(f(x) for x A.
f,g: RR, f(
MAT102: Mathematical Thinking-Lecture 4: Sets
Definition: A set is a collection of distinct objects. The objects are called the elements or the
members of the set.
Notation and terminology:
1. Sets will be labled using uppercase letters (A,B,C,.).
MAT102: Mathematical Thinking-Lecture 3: Elementory Inequalities
Definition: The absolute value of a real number x, written as |x|, is defined as follows:
x x , if x 0
x , if x< 0
We think of
Last / Surname
First / Given Name
MAT 102S - WINTER 2015
QUIZ 4 - SOLUTIONS
Candy Chou, Mon 9-10am (TUT0101).
Mark Zietara, Mon 10-11am (TUT0102).
MAT102S - Introduction to Mathematical Proofs - Winter 2016 - UTM
Problem Set I - TO BE SUBMITTED TO YOUR TA
Monday, March 14, in tutorials.
This assignment must be submitted to your TA at the beginning of the tutorial on the above date.
UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2015 FINAL EXAMINATION
MAT102H5S - Introduction to Proofs
J. Thind, J. De Simoi, A. Mousavidehshikh
Duration: 2 hours
Last / Surname First / Given Name
STUDENT #: SIGNATURE:
MAT102 - Intro. to Mathematical Proofs - Winter 2014 - UTM
Term Test (Version A) - February 26, 2014
Time allotted: 50 minutes.
Aids permitted: None.
Tutorial Section (fill in completely the appropriate circle):