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

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Published on 29 Sep 2015

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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,...}ℕ

Ex. 2 = {...,-3, -2, -1, 0, 1, 2, 3,...}ℤ

○ 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∪

## 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: