Hvad er forskellen mellem en kongruensrelation og en ækvivalensrelation?


Svar 1:

David Joyces svar er godt, men der er en anden definition på kongruensforhold, som jeg har set (Hungerfords Algebra):

Lad G være en monoid med en ækvivalensrelation ~.

~ er en kongruensrelation, hvis

fora,b,c,din[math]G[/math],if[math]a[/math] [math]b[/math]and[math]c[/math] [math]d[/math]then[math]ac[/math] [math]bd.[/math]for a, b, c, d in [math]G[/math], if [math]a [/math]~[math] b[/math] and [math] c [/math]~[math] d[/math] then [math]ac [/math]~[math] bd.[/math]

Thisisusefultodefinenormalsubgroups,andquotientgroupsbecauseG/ isagroupwithabinaryoperationthatrespectsthecongruencerelation.This is useful to define normal subgroups, and quotient groups because G/~ is a group with a binary operation that respects the congruence relation.


Svar 2:

Therearetworelationsknownascongruencerelations.Oneisingeometryandreferstocongruentfigures.Twofiguresarecongruentifthereisarigidmotionthatmovesonetotheother.Theotherisinnumbertheoryandreferstointegerscongruentmodulonwhere[math]n[/math]issomefixedinteger.Twointegersarecongruentmodulo[math]n[/math]iftheirdifferenceisdivisibleby[math]n.[/math]Thissecondcongruencerelationhasbeenextendedtoelementsofaringmoduloanideal.There are two relations known as congruence relations. One is in geometry and refers to congruent figures. Two figures are congruent if there is a rigid motion that moves one to the other. The other is in number theory and refers to integers congruent modulo n where [math]n[/math] is some fixed integer. Two integers are congruent modulo [math]n[/math] if their difference is divisible by [math]n.[/math] This second congruence relation has been extended to elements of a ring modulo an ideal.

Begge disse er ækvivalensrelationer. Der kan også være andre ækvivalensrelationer, der kaldes kongruensrelationer.

For svaret på dit spørgsmål er en kongruensrelation en bestemt ækvivalensrelation, der er kommet til at blive kaldt en kongruensrelation.