Congruence
2020 James B. Wilson
Colorado State University
Homomorphisms
Quotients
Without Algebra...
Equivalence
Function
Partition
Without Algebra...
\(a\equiv a\),
\(a\equiv b\Rightarrow b\equiv a\),
\(a\equiv b,b\equiv c\Rightarrow a\equiv c\)
\(f:A\to B\)
\(A\leftrightarrow \bigsqcup_{b:B} P_b\)
\(P_b=\{a\mid a\equiv b\}\)
\(a\mapsto P_b\), \(a\in P_b\)
\(a\equiv b\pmod{f}\Leftrightarrow f(a)=f(b)\)
With Algebra...
Congruence
Homomorphism
Quotients
Know as: The Fundamental Homomorphism Theorem
With Algebra...
\[\begin{array}{c}a\equiv a'\\b\equiv b'\\ \hline a*b\equiv a'*b'\end{array}\]
\(f(a*b)=f(a)*f(b)\)
\[x\in P_a,y\in P_b\]
\[\Rightarrow x*y\in P_{a*b}\]
\(P_b=\{a\mid a\equiv b\}\)
\(A/_{\equiv}:=\{P_b\mid b\in A\}\)
\(a\mapsto P_b\), \(a\in P_b\)
\(a\equiv b\pmod{\ker f}\Leftrightarrow f(a)=f(b)\)
(Illustrated for binary operations only.)