From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Wed Apr 16 10:24:12 2008
Date:         Wed, 16 Apr 2008 10:10:38 -0400
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191:  Correction to definition of "congruence mod m"
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

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Subject: 191:  Correction to definition of "congruence mod m"
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The definition I gave in class this morning of the equivalence relation
"congruence mod m" should have been:

Background:  Let x,m \in Z.  Then the arithmetic operator "mod" can be
defined as follows:  x mod m = the remainder upon dividing m into x.

Now:

Let m \in Z.
Then equivalent-mod-m =def {(a,b) \in Z | a mod m = b mod m}

So, a is equivalent-mod-m to b (or:  a is congruent to b, modulo m)
    iff
    a mod m = b mod m

Note that equivalence relations can be "reduced" to identities w.r.t.
some property:

a equiv-mod-m b iff the number "a mod m" = the number "b mod m"

the fractional numeral "a/b" is equivalent to the fractional numeral
"c/d" iff the number represented by the fraction a/b = the number represented
by the fraction c/d

We could define two objects to be color-equivalent iff they have the
same color:  a color-equiv b iff a's color = b's color.

And so on.