Introduction to the Sets

Definition of Sets

In everyday life, a set can be matched with a group. In science lessons we are also familiar with the grouping of living things, such as carnivores, herbivores, and omnivores. In social life, we are also familiar with community groupings, such as state, ethnicity, province, district, and so on. All the examples already mentioned above in Mathematics are referred to as Sets. However, not all of these groups can be referred to as sets.

In Mathematics a set is defined as a collection of objects that are clearly defined. A collection of objects or objects is called a member/element of the set.

An object that is a member of the set is denoted by $ \in $, while that which is not a member of the set is denoted by $ \notin $.

Sample writing:

• $\text{mango} \in \text{fruit set}$
• $\text{cat} \notin \text{fruit set}$

Example of a Group that is a Sets

1. A group of four-legged animals.
2. A group of Southeast Asian countries.
3. A set of students born in December.
4. A set of names starting with the letter C.
5. A set of natural numbers.

Example of a Group who is not a Sets

1. A collection of delicious food.
2. A collection of attractive girls.
3. A beautiful collection of flowers.
4. A collection of beautiful paintings.
5. A collection of clever students.

Selection Presentation

1. By naming its members (enumeration)

A set can be stated by naming all its members which are written in curly braces. If the members of the set are very many or infinite, you can put a three dot (...) at the beginning or end.

Example

• $A = \left\{ {1,2,3,4,5} \right\}$
• $B = \left\{ {chicken, bird, duck} \right\}$
• $C = \left\{ { \cdots - 3, - 2, - 1,0,1,2,3, \cdots } \right\}$
• $D = \left\{ {1,3,5, \cdots } \right\}$
2. By specifying the nature of the members

A set can be defined by mentioning the properties of its members.

Example

• A is the set of natural numbers less than $100$.
• B is the set of carnivores.
• C is the set of traffic light colors.
• D is the set of integers between $1$ and $50$.
3. With set-forming notation

A set can be stated by writing down the membership conditions of the set. This notation usually takes the general form $\left\{ {x|P\left( x \right)} \right\}$ where $x$ represents the members of the set, and ${P\left( x \right)}$ states the terms that $x$ must meet in order to be a member of the set. The $x$ symbol can also be replaced with other variables such as $a$, $y$, or others.

Example

• $A = \left\{ {1,2,3,4,5} \right\}$ can be expressed with set-builder notation as $A = \left\{ {x |x \le 5,x \in Original} \right\}$
• $B = \left\{ {2,3,5,7,11} \right\}$ can be expressed by notation set forming $B = \left\{ {y|y < 12,y \in Prima} \right\}$
• $C = \left\{ {red,yellow,green} \right \}$ can be expressed with set-builder notation as $C = \left\{ {x|\text{x is the traffic light color}} \right\}$
• $P = \left\{ {a,i,u,e,o} \right\}$ can be expressed with set-forming notation as $P = \left\{ {y|\text{y is a vowel}} \right\}$

This is an explanation of the definition of a set. If you have any questions, please doodle in the comments column. Hope it's useful