Download UIUC Math 347H Lecture 6: Discussion questions Equivalence

UIUC Math 347H Lecture 6: Discussion questions
Equivalence relations and the construction of Q. We continue
our story of an alien race 10 million light years from here. We guessed
that whatever basic number system they have will satisfy our rules of
arithmetic.
They would eventually worry about that fact that integers are not
closed under division and want to think about a set containing integers
that is closed.
Let
F = {(a, b)|a, b ∈ Z, b 6= 0.
Define an equivalence relation ∼ on F by
(a, b) ∼ (c, d) ifad = bc.
1. Discuss (again) why ∼ is an equivalence relation.
Let {(a, b)} be the equivalence class of the representative (a, b).
2. (Quick sanity check): is {(1, 2)} = {(2, 4)}?
2. TRY to define addition by what you think
{(a, b)} + {(c, d)} = {(ad + bc, bd)},
and multiplication by
{(a, b)} · {(c, d)} = {(ac, bd)}.
What’s the problem?
3. To explain an analogous problem to Q3, consider an equivalence
relation ∼ on all people X by genetic/family relation. TRY to define
{you} > {person next to you} if you know more than the person next
to you about Katy Perry. What’s the problem?
2
4. Prove that multiplication of fractions is well-defined. Now do the
same for addition.
An internal law of composition is a map ◦ : R × R → R. Ex:
+, ×.
Commutative Ring with 1: a set R with +, × satisfying (A1)-(A5),
(M1)-(M4).
If also cancellation (C) ALSO holds (ab = ac and a 6= 0 implies
b = c) then R is an integral domain.
Consider (M5): if a 6= 0 then there is an inverse a−1 such that
aa−1 = a−1 a = 1.
If (M5) ALSO holds, we say R is a field.
5. Let R = Z[x1 , . . . , xn ] be the set of polynomials with integer coefficients. Is (R, +, ×) a commutative ring with 1? An integral domain?
A field?
6. What is the smallest field containing R from Q5?
7. Same question as Q5 except now let R = Matn×n (C).
8. Same question for R = Z10 , Z9 , Z7 . What’s the key difference?
9. What happens if you try to define fractions for R = Z10 ?
10. What about for R = Z7 ?
Go read Section 7. Functions (and specifically the Pigeonhole principle).