User:TeacherTresha/sandbox

SETS
Learning Objectives

After completing this topic, the students are expected to:

1. describe sets using the roster or set builder method and;

2. use and interpret set notation correctly.

DEFINITION
A set is a collection of clearly defined objects which are called elements. Sets are usually denoted by capital letters of the English alphabet while small letters denote elements.

Notations
1. ∈ means ‘is an element of’ or ‘belongs to’ or ‘is a member of’.

2. ∉ means ‘is not an element of’ or ‘does not belong to’ or ‘is not a member of’.

Example 1: The following are sets:
1. A = the set of all consonants of the English alphabet

2. B = the set of all positive integers

3. C = the set of all colleges of BISU Main Campus

4. D = The set of all dogs with 2 legs

5. E = The set of all counting numbers from 1 to 50

It can be seen that in example 1, ℎ ∈ 𝐴, $$1/5 $$ ∉ 𝐵, 𝐶𝑇𝐸 ∈ 𝐶, 𝑆ℎ𝑖ℎ 𝑇𝑧𝑢 ∉ 𝐷,and −8 ∉ 𝐸.

Roster Method
– lists, enumerates, or tabulates distinct elements of a set which are separated by a comma and enclosed by a pair of braces { }.

Example 2: The sets in example 1 can be described using roster method as
𝐴 = {𝑏, 𝑐, 𝑑, 𝑓, 𝑔, ℎ,𝑗, 𝑘, 𝑙, 𝑚, 𝑛, 𝑝, 𝑞, 𝑟, 𝑠,𝑡, 𝑣, 𝑤, 𝑥, 𝑦, 𝑧}, 𝐵 = {1,2,3,4,5, … },

𝐶 = {𝐶𝑇𝐸, 𝐶𝐸𝐴, 𝐶𝑇𝐴𝑆, 𝐶𝐵𝐴𝑆, 𝐶𝐴𝑑𝑆}, 𝐷 = { } or 𝐷 = ∅ and

𝐸 = {1,2,3,4,5, … ,49, 50}.

Note that the order of listing the elements of a set does not affect the set itself. The three dots (… ) in set 𝐵 means there are still positive integers after 5 that belong to set 𝐵 while there are still counting numbers between 5 and 49 that belong to set 𝐸. Furthermore, a set should have distinct elements.

Rule/Set-builder Method
– uses a phrase that describes the property of the elements of the set and enclosing it by a pair of braces The rule method uses the set descriptive notation {𝑥: 𝑥 is a (descriptive phrase)}. The letter 𝑥 is the variable and is used to represent any element of a set and maybe replaced by any other letter. Also the vertical bar symbol, “|” maybe used instead of colon “:” and is read as such that.

Example 3: The sets in the previous example can be described using rule method as

𝐴 = {𝑥: 𝑥 is a consonant of the English alphabet}

𝐵 = {𝑦: 𝑦 is a positive integer}

𝐶 = {𝑧: 𝑧 is a college of BISU Main campus}

𝐷 = {𝑥|𝑥 is a dog with 2 legs}

𝐸 = {𝑦|𝑦 is a counting number from 1 to 50}

Definition
An empty set or null set, denoted by { } or ∅ is a set that does not contain any element. The set 𝐷 in example 1 is empty.

Definition
A universal set, denoted by 𝑈 is a set containing all elements included in a single discussion. It is the set from which all other sets are derived.

Example 4:
The set of integers ℤ = {…, −3, −2, −1, 0, 1, 2, 3, … }, maybe considered as a universal set of the sets 𝐵 and 𝐸 in example 1.

I. List all the elements of the following sets:
1. 𝐿 = {𝑧 ∈ ℝ | 𝑧 2 − 3𝑧 = 4}

2. 𝑂 = {𝑥 ∈ ℤ |(𝑥 − 1)(𝑥 + 3)(2𝑥 + 3)(𝑥 + 5) = 0}

3. 𝑉 = {𝑦 ∈ ℤ | − 3 ≤ 𝑦 < 3}

4. 𝐸 = {𝑤 ∈ ℕ|𝑤 is an odd single − digit number}

ANSWER

Quilaton

1. L = (-1,4)

2. O = (-5,-3,1)

3. V = (-3,-2,-1,0,1,2)

4. E = (1,3,5,7,9)

Marquita

1. L = { -1,4}

2. O = { -5,-3,1}

3. V = { -3,-2,-1,0,1,2}

4. E = { 1,3,5,7,9}

Cruza

1. L = { -1, 4}

2. O = { -5, -3, 1}

3. V = { -3, -2, -1, 0, 1, 2}

4. E = { 1, 3, 5, 7 ,9}

Arcayera

1. L = { -1, 4}

2. O = { -5, -3, 1}

3. V = { -3, -2, -1, 0, 1, 2}

4. E = { 1, 3, 5, 7 ,9}

II. Describe the following sets, using set-builder notation:
1. 𝑀 = {3,6,9,12, … }

2. 𝐸 = {−3, −2, −1,1,2,3,4,5,6}

ANSWER

QUILATON

1. M = ( x:x is a natural number multiple of 3)

2. E = ( y:y is an integer, -3 ≤ y < 7 where as y ≠ 0)

MARQUITA

1. M= {x:x is a natural number multiple of 3}

2. E= {y:y is an integer,-3 ≤ y < 7 where as y≠0}

CRUZA

1. M= {x:x is a natural number multiple of 3}

2. E= {y:y is an integer, -3 ≤ y < 7 where as y≠0}

ARCAYERA

1. M= {x:x is a natural number multiple of 3}

2. E= {y:y is an integer, -3 ≤ y < 7 where as y≠0}