Lesson Objectives:

Dear Learners !

At the end of the lesson, you will be able to:

• Define Key Concepts.
• Identify and Use Notation.
• Apply Membership Symbols.
• Describe Sets Using Various Methods
• Apply the Concept of Sets to Mathematical and Real-World Contexts

Key Terms

• Set
• Member
• Weldefindness
• Null set

Brainstorming Questions

1: Consider all of your family members (your mother, father, sister, brother, and others) who live in your home, and state the names of each member, including yourself, in a set. (What do we call this?)

2: Consider the set of all students in a classroom who have a birthday in January. Is this a well-defined set? Why or why not?

3: Imagine a set consisting of “the best movies of all time.” Would this be considered a well-defined set? Explain your reasoning.

Answer for Question 1: It is a set.

Answer for Question 2: Yes, this is a well-defined set because the criterion for membership (having a birthday in January) is clear and unambiguous. Every student either has a birthday in January or they do not, making it easy to determine the elements of the set.

Answer for Question 3: No, this would not be a well-defined set because the criterion for what makes a movie “the best” is subjective and varies among individuals. Different people have different opinions on what constitutes the best movies, leading to ambiguity and lack of consensus.

1.1.Sets and Elements

A set is a collection of well-defined objects or elements. “well-defined” indicates that there is a consensus among individuals decision regarding the elements of the set, with no ambiguity or variation in interpretation.

Example 1:

Consider the set of all books written by J.K. Rowling. Is this a well-defined set? Why or why not?

Solution:

Yes, this is a well-defined set because the criterion for membership (being written by J.K. Rowling) is clear and unambiguous. A book either is written by J.K. Rowling or it is not, making it easy to determine the elements of the set.

But, The set of good students in Ethiopia grade nine students, the set of beautiful girls in Ethiopia, the set of colorful birds in your area, and others are not a set. Because, there are variations in decisions of individuals to select the element of a set or there are ambiguity or variation in interpretation.

Note:

• Sets usually denoted by by capital letters like 𝐴, 𝐵, 𝐶, 𝑋, 𝑌, 𝑍, etc.

• Elements or objects of a set represented by small letters like 𝑎, 𝑏, 𝑐, 𝑥, 𝑦, 𝑧, etc are separated by commas with in curly braces {}.
• The Greek symbol ∈ (epsilon) is utilized to signify “belongs to” or “is an element of”.

Consider a set $A = \{a, b, c, d, e, f, x, z\}$.

Hence, we express $a \epsilon A$ to indicate that 𝑎 is a part of set 𝐴. Conversely, if y is not a member of set 𝐴, we denote this as y ∉ 𝑨 and interpret it as “y does not belong to set 𝐴” or “y is not a member of set A”.

Example 2:

For the set of positive even numbers A={2, 4, 6, 8, … . }, we have 4 ∈ 𝐴, 5 ∉ 𝐴, −2 ∉ 𝐴, 0 ∉ 𝐴, 12 ∈ 𝐴.

1.2. Description of sets

Sets can be defined using various methods. These include: 

i. Verbal method (Statement form):

Sets can be described in words, stating the characteristics or properties of the elements in the set. Example: The set of natural numbers less than 1000.

ii. Listing Method

a. Complete listing method (Roster Method): In this method, all the elements of the set are explicitly listed within curly braces {}.

Example 3: $A =\{ 1, 2, 5, 11, 19, 23\}$

b. Partial listing method: In this method, only some elements of the set are listed, typically followed by ellipses (…) to indicate that the pattern continues in the same pattern.

Example 4: $B = \{2, 4, 6, 8, 10, 12, \dots\}$.

iii. Set builder method (Method of defining property):

In this method, a set is defined by specifying a property or condition that its elements must satisfy. The set is represented as $\{x | P(x)\}$, where x is the variable representing the elements of the set, and $P(x)$ is the property that determines whether x belongs to the set.

Example 5: $A = \{ x \mid x \in \mathbb{N}, x < 21 \}$

In conclusion, We have discussed sets, elements and different methods of defining sets, such as verbal statements, listing methods, and set builder methods. Sets allow mathematicians to study relationships between objects, perform operations, and develop mathematical theories across various disciplines

Basic concepts for Sets and Elements

• Element: An individual object or member of a set. Elements are often denoted by lowercase letters.
• Well-defined: A characteristic of a set where the criteria for inclusion in the set are clear and agreed upon, leaving no room for ambiguity.
• Roster Method (Listing Method): A method of defining a set by explicitly listing all of its elements within curly braces. Example: A = {a, b, c} .
• Partial Listing Method: A variation of the roster method where only some elements of the set are listed, followed by ellipses (…) to indicate that the pattern continues. Example: B = {2, 4, 6, …} .
• Set Builder Notation (Set Builder Method): A method of defining a set by specifying a property that its elements must satisfy. Represented as ${x \mid P(x)} $, where $ x $ is a variable representing elements and $ P(x) $ is the condition for membership. Example: $ C = \{ x \mid x \text{ is a positive integer less than } 10 \}$.
• Membership: The relationship between an element and a set. Denoted by $( \in ) (epsilon)$ for “belongs to” and $( \notin )$ for “does not belong to”. Example: $( a \in A )$ means $a $ is an element of set $ A $.
• Non-Well-defined Set: For Example:- The set of good students in Ethiopia grade nine students.
• Complete Listing: A = {1, 2, 3, 4, 5}.
• Partial Listing: $( B = {2, 4, 6, 8, \ldots} )$ (where the pattern continues).

Conclusion

Sets and elements are fundamental concepts in mathematics that allow for the organization and classification of objects based on well-defined criteria. Understanding these concepts is crucial for studying relationships, performing operations, and developing mathematical theories.