MATH1050A Lecture Outline 14
0. Objective: Image sets and pre-image sets.
1. Definition (image sets). Let A, B be sets and f : A B be a function from A to B. Let S be a subset of A. The
image set of the set S under the function f is defin
MATH1050A Lecture Outline 10
0. Objective: Universal quantifier, existential quantifier, statements with quantifiers.
Going through the mathematical statements in your mathematics textbooks very carefully, we will find that the
MATH1050A Lecture Outline 9
0. Objective: Calculus of logic, logical equivalence, rules of inference.
Questions. What make the proofs we have seen work? How do we know we are logically correct when moving from
one line to the next in a pro
MATH1050A Lecture Outline 12
0. Objective: Surjectivity and injectivity.
1. Notion of surjectivity.
Definition (surjectivity). Let A, B be sets, and f : A B be a function from A to B. f is said to be surjective if
the following statement
MATH1050 Handout: Compositions, Surjectivity and Injectivity
Statement (1). Let A, B, C be sets, and f : A B, g : B C be functions. Suppose f, g are
surjective. Then g f is surjective.
Proof 1 (with pictures). Let A, B, C be sets, and f : A B, g : B C be
MATH1050 Handout: Results involving image sets and pre-image sets
Let A, B be sets, and f : A B be a function. The following statements hold:
(1a) f () = , f 1 () = ; f (A) B, f 1 (B) = A.
(1b) For any x A, f (cfw_x) = cfw_f (x).
(1c) For any
MATH1050 Handout: Image Sets and Pre-image Sets
Definition of image sets. Let A, B be sets and f : A B be a function from A to B. Let S be
a subset of A. The image set of the set S under the function f is defined to be the set
cfw_y B : There exists some
MATH1050 Handout: Styles of Direct Proofs
Definition. Let A, B be sets.
The intersection of the sets A, B is defined to be the set cfw_x | x A and x B. It is denoted by
The union of the sets A, B is defined to be the set cfw_x | x A or x B. It is d
MATH1050 Handout: Basic Set Operations
Definition. Let A, B be sets. We say A is equal to B if both of the following statements hold:
For any object x, [if (x A) then (x B)].
For any object x, [if (x B) then (x A)].
We write A = B.
Definition. Let A, B
MATH1050A Lecture Outline 7
0. Objective: Cartesian product, ordered pair.
We introduce a set operation which is analogous to forming the (coordinate) plane from the (number) line.
1. Definition of coordinate pair. (Generalization from sch
MATH1050A Lecture Outline 4
0. Objective: proofs by contradiction; subsets.
1. Proofs by contradiction.
To demonstrate that a statement is true, we sometimes proceed as described in (1) or (2):
(1) In case the statement is very simple, wit
MATH1050A Lecture Outline 16
0. Objective: More on pre-image sets, image sets and surjectivity, injectivity.
Recall definitions of image sets, pre-image sets. Recall Theorem (1), Theorem (2).
1. Theorem (3). Let A, B be sets, and f : A B
MATH1050 Handout: Surjectivity and Injectivity
Definition of surjectivity. Let A, B be sets, and f : A B be a function from A to B. f is said
to be surjective if the following statement holds:
(S): For any y B, there exists some x A such that y = f (x).
MATH1050A Lecture Outline 2
0. Objective: Set notations, basic set operations.
Question. When do we really need set language in mathematics?
What is wrong with 0, 1, 4, 9, 16, ? What is ? Do we really mean the entirety of all square integer
MATH1050A Lecture Outline 3
0. Objective: direct proof.
In order to study a mathematical problem, mathematicians formulate what they suspect to be true or to be false
into mathematical statements, and then they try to prove or dis-prove the
MATH1050A Lecture Outline 17
0. Objective: Notion of inverse function of a function. Bijective functions.
1. Inverse function of a function.
Definition (inverse function). Let A, B be sets, and f : A B, g : B A be functions. g is said to
MATH1050A Lecture Outline 1
What this course is about: two meanings of foundation of modern mathematics. (1) Foundation of modern
mathematics in its literal sense: the language of logic, sets and functions. (2) Foundation f
MATH1050A Lecture Outline 5
0. Question. What is the Principle of Mathematical Induction?
1. Usual formulation of Principle of Mathematical Induction (UPMI). Let P (n) be a predicate, where n is a variable.
Suppose the statement P (0) is t
MATH1050A Lecture Outline 11
0. Objective: Function as assignment from a set to a set. Composition of functions.
Question. How did/do we define a real-valued function of one real variable? [Reserve space for blackboard trick for
MATH1050 Handout: Characterizations of Surjectivity and Injectivity
Theorem (). Let A, B be sets and f : A B be functions. f is surjective iff (for any subset U of
B, f (f 1 (U ) U ).
Let A, B be sets and f : A B be functions.
Suppose that f is su
MATH1050 Handout: More on styles of Direct Proofs
Definition. Let m, n Z. m is said to be divisible by n if there exists some k Z such that m = kn.
Theorem (Properties of divisibility). The following statements hold:
(a) Let x Z. x is divisible by x.
MATH1050 Handout: Notion of Inverse Functions
Definition of inverse function. Let A, B be sets, and f : A B, g : B A be functions. g is
said to be an inverse function of f if both of the following statements hold:
(g f )(x) = x for any x A.
MATH1050A Lecture Outline 8
0. Objective: power set as a special set operation for creating larger and larger sets; dis-proof by counter-examples;
iff, necessary conditions, sufficient conditions.
1. Recall definition: Let A be a set. The
MATH1050 Handout: Principle of Mathematical Induction
Equivalent formulations of the Principle of Mathematical Induction:
1. Let P (n) be a predicate, where n is a variable.
Suppose the statement P (0) is true.
Further suppose that for any k N, if the sta
MATH1050A Lecture Outline 13
0. Objective: Composition of functions, surjectivity and injectivity.
1. Theorem (1 ). Let A, B, C be sets, and f : A B, g : B C be functions. The following statements hold:
(1) Suppose f, g are surjective. Th
MATH1050A Lecture Outline 6
0. Objective: Well-ordering Principle for Integers and Division Algorithm (for Natural Numbers). Existence, uniqueness, existence-and-uniqueness.
1. Definition (least element in a set of numbers). Let S be a sub
MATH1050 Handout: Pictorial visualization of the notion of functions
In-formal definition of function.
Let A, B be sets. A function from A to B is a rule of assignment from A to B, so that each
element of A is being assigned to exactly one element of B.
MATH1050A Lecture Outline 15
0. Objective: Theoretical results about image sets, pre-image sets.
Recall definition of image sets, pre-image sets.
1. Theorem (1). Let A, B be sets, and f : A B be a function. The following statements hold:
MATH1050A Further Exercise 1 (Solution)
1. (a) Let a, b R. Suppose the roots of the quadratic polynomial x2 + ax + b are even integers.
The roots of x2 + ax + b are 2k, 2 for some k, Z.
We have x2 + ax + b (x 2k)(x 2) x 2(k + )x + 4k.
Then a = 2(k + ) and
MATH1050A Further Exercise 3
Due date: 5-10-2014
Questions which require more thought and/or work and/or tricks and/or organization and/or . are marked by
, , , , in ascending order of overall difficulty level.
1. Consider each of the sets below. List eve
MATH1050A Further Exercise 2 (Solution)
1. (a) Denote by P (n) the proposition
+ 1 + 2 + + n = 2 n .
=0=2 0 .
Then P (0) is true.
+ 1 + 2 + + k = 2 k .
We are going to prove that P (k + 1) is true.
MATH1050A Further Exercise 3 (Solution)
1. (a) The elements of A are 0, 1, 2, 3.
(b) The elements of B are 0, 1, 2, 3, 4.
(c) The elements of C are 0, 1, 2, cfw_0, cfw_1.
(d) The elements of D are 0, 1, cfw_0, 1.
(e) The elements of E are 0, 1, cfw_0, 1,
MATH1050A Tutorial 5 (Solution)
Let A, B be sets and f : A B be a function from A to B.
Let S be a subset of A. The image set of the subset S under the function f is defined to be the
set cfw_y B : y = f (x) for some x S . It is denoted by f (S).