Sets and Set Notations Class Notes

Methods of Describing Sets and Set Notations

The lesson on Sets and Set Notations, referenced from the document T MATHS 9-1-2026.pdf and dated Wednesday, January 2086, details the fundamental ways to define a mathematical set. There are three primary methods discussed for describing a set. The first method is describing a set in words; for example, set $S$ is defined as the even positive (tve) integers less than $10$. This approach uses natural language to set the parameters for membership within the collection.

The second method involves listing all numbers or elements of a set within a standard set notation. In this instance, using the previous criteria, the set is written as $S = \{2, 4, 6, 8, 10\}$. It is noted that although the verbal description specifies integers less than $10$, the listed example includes the number $10$. This method provides an explicit and exhaustive list of every member belonging to the set, enclosed within braces.

The third method is describing the elements of a set using set-builder notation. This is represented as $S = \{x \mid x \text{ is a positive even integers less than 10}\}$. The instructional text explains that this notation is read as "$x$ is such that $x$ is positive even integers, less than $10$." This formal method uses a variable $x$ to represent any element in the set and provides a logical predicate that those elements must satisfy.

Theoretical Definitions of Subsets

A subset is identified by the relationship between two sets where one is entirely contained within the other. According to the lesson, a subset exists when every element of one set, labeled $A$, is also present in another set, labeled $B$. The transcript specifies that a subset is "$A$ des in $B$ of every elemenil because it's name. I also present in $B$." This indicates that membership in set $A$ automatically implies membership in set $B$.

An example is provided to illustrate this concept: let set $A = \{1, 2, 3\}$ and set $B = \{1, 2, 3\}$. Since every set of $A$ (notated in the text with the reference $2-5\,2$) is also found in $B$, set $A$ is considered a subset of $B$. The mathematical notation used to denote this relationship is $A \subset B$. This signifies that set $A$ is a part of or equal to set $B$, following the rule that no element exists in $A$ that is not also in $B$.

Characteristics and Notation of Supersets

The concept of a super set (notated as $B \supseteq A$) is the converse of the subset relationship. The lesson defines a set $B$ as a super set of another set $A$ if every element of $A$ is also contained within set $B$. This definition mirrors the subset logic but shifts the perspective to the larger, encompassing set. If set $B$ contains all elements that are in $A$, then $B$ is the super set of $A$.

The formal notation for a super set is denoted by the symbol $B \supseteq A$. The transcript emphasizes that the presence of every element is the defining characteristic of this designation. This means that for set $B$ to qualify as a super set, there must be no element in set $A$ that is missing from set $B$, effectively making $B$ the container for the entirety of set $A$.