Sets

A set is a collection of distinct objects, called elements (or members). They are typically denoted by a capital letter (such as $A, B, C,$ etc.), and their elements can be written inside curly brackets. For example:

\[A=\{1, 2, 3, 4, e, 9.\overline9, 8086 \}\]

Here, $1 \in A$ denotes that “$1$ is an element of $A$”, and $5 \notin A$ means that “$5$ is not an element of $A$”.

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Types of Sets

Name	Description	Example
Finite set	Contains a finite number of elements.	$B={1, 2, 3}$
Infinite set	Contains an infinite number of elements	$\mathbb{N}= {1, 2, 3, \dots}$
Singleton set	Contains exactly one element.	$C={256}$
Universal set	Contains all objects under consideration in a certain context. Denoted as $U$.
Empty set	(Sometimes called the null set) Contains no elements, and is denoted as $\varnothing$ or ${}$

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Subsets

A set $A$ is considered a subset of $B$ if every element in $A$ is also an element of $B$. In which case:

\[A \subseteq B\]

which means: “$A$ is a subset of or equal to $B$”.

If $A$ is a subset of $B$, but $A \neq B$, then $A$ is considered a proper subset of $B$, shown as:

\[A \subset B\]

The empty set ($\varnothing$) is a subset of every set, including singletons, since it has no elements.