# MATH135 Lecture Notes - Lecture 5: Empty Set, Asteroid Family, Cartesian Coordinate System

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MATH 135 - Lecture 5: More Divisibility and Introduction to Sets
Divisibility of Integer Combinations (DIC):
Let a, b, and c be integers. If a | b and a | c then for any integers x and y, a | (bx + cy).
Proof with numbers:
Implication: If m and 14 | ∈ ℤ m, then 7 | (135m + 693)
Proof: Assume m and 14 | ∈ ℤ m.
Since 14 | m, therefore 7 | m.
Since 693 = 99(7), therefore 7 | 693
Therefore 7 | (135m + 693) by DIC.
Sets:
A well-defined collection of objects
Objects that make up a set are called elements
Ex. A = {2, 4, 6, 8}
Note: Order in a set does not matter. Repetition does not matter
2 A → 2 belongs to set A, 3 A → 3 does not belong to set A∈ ∉
We can represent a set in different ways:
a complete list → A = {2, 4, 6, 8}
a list using “...” → B = {2, 4, 6, 8,…,100}
Ex. 1 = {1, 2, 3, 4,...}
Ex. 2 = {...,-3, -2, -1, 0, 1, 2, 3,...}
using set-builder notation → A = {x : ∈ ℕ x is even 2 ≤ x ≤ 8}
The smallest possible set is the empty set: = { }ϕ
Note: { } this is a set that contains the empty set and thus is not emptyϕ ϕ →
The largest possible set for a given situation is called the universal set or the universe of
discourse: usually notated by the uppercase letter U.
Ex. Describe the following sets using set-builder notation
1. All multiples of 7
{x : 7 | ∈ ℤ x}
2. All odd perfect squares
{n2 : n = 2k + 1 for some k }∈ ℤ
3. All points on a circle of radius 8 centred at the origin
{{x, y) : (x, y ) (∈ ℝ x2 + y2 = 64)}
4. All sets of three integers which are the side lengths of a non-trivial triangle
{(x, y, z) : (x, y, z ) (∈ ℕ x < y + z) (y < x + z) (z < x + y)}
Set Operations:
the cardinality of a set S refers to the number of elements in a finite set, denoted |S|
Ex. for A = {2, 4, 6, 8}, |A| = 4
the union of two sets: S T
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## Document Summary

Math 135 - lecture 5: more divisibility and introduction to sets. Let a, b, and c be integers. If a | b and a | c then for any integers x and y, a | (bx + cy). Proof: assume m and 14 | and 14 | m, then 7 | (135m + 693) m. Since 14 | m, therefore 7 | m. Therefore 7 | (135m + 693) by dic. Objects that make up a set are called elements. Note: order in a set does not matter. A 3 does not belong to set a. A 2 belongs to set a, 3. We can represent a set in different ways: A complete list a = {2, 4, 6, 8} A list using b = {2, 4, 6, 8, ,100} = {,-3, -2, -1, 0, 1, 2, 3,} Using set-builder notation a = {x. The smallest possible set is the empty set: