Vad är skillnaden mellan en kongruensrelation och en ekvivalensrelation?


Svar 1:

David Joyces svar är bra, men det finns en annan definition för kongruensrelation som jag har sett (Hungerford's Algebra):

Låt G vara en monoid med en ekvivalensrelation ~.

~ är en kongruensrelation om

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.

Båda dessa är ekvivalensrelationer. Det kan också finnas andra ekvivalensrelationer som kallas kongruensrelationer.

För svaret på din fråga är en kongruensrelation en speciell ekvivalensrelation som har kommit att kallas en kongruensrelation.