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+#+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
+
+