Last
edited 5/9/2007 8:41:00 PM
QUOTIENT AND IMAGE
131. The kernel equivalence of
a function
If F : A - B is a function, it induces an equivalence relation K(F)
on its domain A by identifying elements that go to the same
thing in B. Formally:
131.1 Definition: kernel equivalence
If F: A - B is
a function, the kernel equivalence of F on A, denoted K(F), is defined by
aK(F)a' 
F(a) = F(a')
131.1.1 Fact It is easy to see that the kernel
equivalence of a function is an equivalence relation.
131.1.2 Example The congruence relations described in the
preceding section are kernel equivalences. Let k be a
fixed integer ≥ 2. The remainder function F: Z -
Z is defined by F(n) = n (mod k), the remainder
when n is divided by k. Theorem
130.2, reworded, says exactly that the relation of congruence (mod k) is
the kernel equivalence of the remainder function.