#+TITLE: Sample Problems

WELL-DEFINED:
when x \in domain = y \in domain then f(x) = f(y) and each x \in domain has y \in range

PRINCIPAL IDEAL:
ideal generated by an element

SURJECTION:
every y \in range has (non-unique) x such that f(x) = y

* Question Three
** a
$f$ is well defined as $f([a]_12)$ = $f([b]_12)$ implies $[a] = [b]$, as $[a]_4 = [b]_4$ implies that $a \equiv_4 b$ so $a - b \equiv_4 0 \Rightarrow a - b \equiv_4 12$
which implies $a - b = 12n$
** b
$f$ is a homomorphism:
f(a + b) = f(a) + f(b) by f([a]_12 + [b]_12) = f([a + b]_12) = [a + b]_4 = [a]_4 + [b]_4 = f([a]_12) + f([b]_12)
f(ab) = f(a)f(b) by f([a]_12 [b]_12) = f([ab]_12) = [ab]_4 = [a]_4[b]_4 = f([a]_12)f([b]_12)

$f$ is surjective:
every element in the range [a]_4 can be mapped to an element in the domain: [a]_12 since f([a]_12) = [a]_4
** c
the kernel are all the elements of $Z_12 \equiv_4 0$ (0, 4, 8)

** d
first isomorphism theorem: it's Z_4

* Question Four