Common Symbols Used in Set Theory

Symbols save time and space when writing. Here are the most common set symbols

Reference: https://www.mathsisfun.com/sets/symbols.html

In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}

Symbol	Meaning	Example
{ }	Set: a collection of elements	{1, 2, 3, 4}
A $∪$ B	Union: in A or B (or both)	C ∪ D = {1, 2, 3, 4, 5}
A $∩$ B	Intersection: in both A and B	C ∩ D = {3, 4}
A $⊆$ B	Subset: every element of A is in B.	{3, 4, 5} ⊆ D
A $⊂$ B	Proper Subset: every element of A is in B, but B must have more elements.	{3, 5} ⊂ D
A $⊄$ B	Not a Subset: A is not a subset of B	{1, 6} ⊄ C
A $⊇$ B	Superset: A has the same elements as B, or more	{1, 2, 3} ⊇ {1, 2, 3}
A $⊃$ B	Proper Superset: A has B's elements and more	{1, 2, 3, 4} ⊃ {1, 2, 3}
A $⊅$ B	Not a Superset: A is not a superset of B	{1, 2, 6} ⊅ {1, 9}
$A^c$	Complement: elements not in A	$D^c$ = {1, 2, 6, 7} When $\mathbb{U}^c$ = {1, 2, 3, 4, 5, 6, 7}
A − B	Difference: in A but not in B	{1, 2, 3, 4} − {3, 4} = {1, 2}

a $∈$ A	Element of: a is in A	3 ∈ {1, 2, 3, 4}
b $∉$ A	Not an element of: b is not in A	6 ∉ {1, 2, 3, 4}
Ø	Empty set = {}	{1, 2} ∩ {3, 4} = Ø
$\mathbb{U}$	Universal Set: set of all possible values (in the area of interest)

P(A)	Power Set: all subsets of A	P({1, 2}) = { {}, {1}, {2}, {1, 2} }
A = B	Equality: both sets have the same members	{3, 4, 5} = {5, 3, 4}
A×B	Cartesian Product(set of ordered pairs from A and B)	{1, 2} × {3, 4}= {(1, 3), (1, 4), (2, 3), (2, 4)}
\|A\|	Cardinality: the number of elements of set A	\|{3, 4}\| = 2
\|	Such that	{ n \| n > 0 } = {1, 2, 3,...}
$:$	Such that	{ n : n > 0 } = {1, 2, 3,...}
$∀$	For All	∀x>1, x2>xFor all x greater than 1x-squared is greater than x
$∃$	There Exists	∃ x \| x2>xThere exists x such thatx-squared is greater than x
∴	Therefore	a=b ∴ b=a

$\mathbb{N}$	Natural Numbers	{1, 2, 3,...} or {0, 1, 2, 3,...}
$\mathbb{Z}$	Integers	{..., −3, −2, −1, 0, 1, 2, 3, ...}
$ℝ$	Rational Numbers	{…., 1/2, 3/4, -2/5, 0, 2, -7, …}
$\mathbb{R}$	Real Numbers	{…, 1.5, -√2, 0.25, π (pi), 2, -5, …}
$\mathbb{I}$	Imaginary Numbers	{…, 2i, -3i, 0.5i, 6i, …}
$\mathbb{C}$	Complex Numbers	{…, 3 + 2i, -1 - 4i, 0 + 7i, 5 - 2i, …}

Concepts:

To study set theory effectively, it is recommended to cover the following topics in a logical order:

Basics of Sets: Understand the fundamental concepts of sets, such as elements, subsets, universal set, empty set, and set notation.
Set Operations: Learn about various operations on sets, including union, intersection, and complement. Study the properties and relationships among these operations.
Venn Diagrams: Explore the graphical representation of sets using Venn diagrams. Understand how to represent set relationships and set operations visually.
Cardinality of Sets: Introduce the concept of cardinality, which refers to the number of elements in a set. Study different types of sets, such as finite, countably infinite, and uncountable sets.
Power Set: Learn about the power set of a set, which is the set of all possible subsets of a given set. Understand its cardinality and properties.
Cartesian Product: Explore the concept of the Cartesian product of sets. Understand how it is used to define relations and functions.
Equivalence Relations: Study equivalence relations and their properties, including reflexivity, symmetry, and transitivity. Learn about equivalence classes and partitions of sets.
Order Relations: Introduce order relations, such as partial order and total order. Study related concepts like well-ordering, minimal and maximal elements, and linear extensions.
Functions and Relations: Deepen the understanding of functions and relations between sets. Study injectivity, surjectivity, and bijectivity of functions. Explore different types of relations, including equivalence relations and partial orders.
Axiomatic Set Theory: Familiarize yourself with the axioms of set theory, such as Zermelo-Fraenkel (ZF) axioms, and their implications. Study the construction of the natural numbers, ordinal numbers, and cardinal numbers within set theory.