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Set Theory

Jul 17, 2023

Abstract Algebra

# Basic Sets

N := Natural Number
Z := Integer
Q := Rational Number
R := Real Number
C := Complex Number

# Term

# Disjoint

When two sets have no elements in common, they are said to be disjoint

# Complement

$A ^ { \prime } = { x : x \in U \text{ and } x \notin A }$

# Difference

$A \backslash B = A \cap B ^ { \prime } = { x : x \in A \text{ and } x \notin b}$

# Relations

Subsets of $A × B$ are called relations

A relation is well-defined if each element in the domain is assigned to a unique element in the range.

# Surjective or onto

each element in B has an A

# Injective or One-to-one

$a_1 \neq a_2 \Leftrightarrow f(a_1) \neq f(a_2)$

# Bijective

A map that is both one-to-one and onto is called bijective.

# Identity Mapping

If S is any set, we will use ids or id to denote the identity mapping from S to itself.
Define this map by id(s)= s for all s $\in$ S.

# Invertible

A mapping is invertible if and only if it is both one-to-one and onto.

Equivalence relation

# Manipulation