What have we learned in Algebra?
Matrix
- Equivalence
- Similarity
- Congruence
Polynomial
- Univariate polynomial
- Multivariate polynomial
- Characteristic polynomial
Set theory and Maps
Relationship
- $\in$, $\notin$
- $\subset$, $\subseteq$, $\subsetneq$, $\nsubseteq$
- $A=B \Leftrightarrow A \subset B, B \subset A$
Operation Symbol
- $A\bigcup B$
- $A \bigcap B$
- $A^c \sim \overline{A} \sim I \setminus A$
- $A \setminus B \sim A \bigcap \overline{B}$
- $A \oplus B \sim (A \setminus B) \bigcup (B \setminus A)$
Maps
- $f: X \rightarrow Y$
- Image : $Im(f)={f(x)| x \in X }$
- Graph : $Graph(f)={ ( x , f(x) ) | x \in X }$
Equivalence Relation
Definition
An equivalence relation on a set $S$ is a set $R$ of ordered pairs of elements of $S$ such that
- $(a,a)\in R$ for all $a \in S$ ( reflexive property ).
- $(a,b) \in R$ implies $(b,a)\in R$ ( symmetric property ).
- $(a,b) \in R$ and $(b,c)\in R$ imply $(a,c)\in R$ ( transitive property ).
When $R$ is an equivalence relation on a set S, it is customary to write $aRb$ instead of $(a,b)\in R$. If $\sim$ is an equivalence relation on a set $S$ and $a \in S$, then the set $[a]={ x \in S| x \sim a }$ is called the equivalence class of $S$ containing $a$.
Examples
- Example 1 Let $S$ be the set of all triangles in a plane. If $a,b \in S$, define $a \sim b$ if $a$ and $b$ are similar —— that is, if $a$ and $b$ have corresponding angles that are the same. Then $\sim$ is an equivalence relation on $S$.
- Example 2 Let $S$ be the set of all polynomials with real coefficients. If $f$, $g\in S$, define $f \sim g$ if $f^\prime=g^\prime$. We see that for any $f$ in $S$, $[f]={ f+c|c\mathrm{ \; is \; real } }$.
- Example 3 Let $S$ be the set of integers and let $n$ be a positive integer. If $a$, $b\in S$, define $a\equiv b$ if $a\pmod{n}=b$. Then $\equiv$ is and equivalence relation on $S$ and $[a]={a+kn|k\in S}$.
- Example 4 Let $S={(a,b)|a,b \;\mathrm{ are\; integers,}\; b \neq 0}$. If $(a,b),(c,d)\in S$, define $(a,b) \approx (c,d)$ if $ad=bc$. Then $\approx$ is an equivalence relation on $S$. [ The motivation for this example comes from fractions. In fact, the pairs $(a,b)$ and $(c,d)$ are equivalent if the fractions $a/b$ and $c/d$ are equal. ]
- Example 5 Any map of sets $f: S \rightarrow T$ gives us an equivalence relation on its domain $S$. $R$ is defined by the rule $a\sim b$ if $f(a)=f(b)$.
- Example 6 If G is a finite group, we can define a map $f:G\rightarrow\mathbb{N}$ to the set { 1, 2, 3, … } of natural numbers, letting $f(a)$ be the order of the element $a$ of $G$. the fibres of this map are the sets of elements with the same order.