# Assignment 1

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10 Nov 2012
School
Department
Course
Professor
Assignment 1
MAT 157
Due: September 19, 2012
1. Consider the set
M={
a b
c d
:a, b, c, d R}
with the usual operations of addition and multiplication:
a b
c d
+
a0b0
c0d0
=
a+a0b+b0
c+c0d+d0
and
a b
c d
·
a0b0
c0d0
=
aa0+bc0ab0+bd0
ca0+dc0cb0+dd0
.
Deﬁne the new operation
[X, Y ] = XY Y X.
Prove that
(a) For all X, Y ∈ M, we have [X, Y ] = [Y, X] and in particular, the operation
is not commutative.
(b) There exist elements X, Y, Z ∈ M such that
[[X, Y ], Z]6= [X, [Y, Z]]
and so the operation is not associative.
(c) Prove that for all X, Y, Z ∈ M, we have the identity
[X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0.
2. (Spivak, Chapter 1, Problem 7) Prove that if 0 < a < b, then
a <
ab <
1
2
(a+b)< b.
1
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## Document Summary

Due: september 19, 2012: consider the set with the usual operations of addition and multiplication: M = { c d (cid:18) a b (cid:18) a(cid:48) (cid:19) (cid:18) a(cid:48) b(cid:48) c(cid:48) d(cid:48) (cid:19) b(cid:48) c(cid:48) d(cid:48) (cid:19) (cid:19) (cid:18) a b (cid:19) (cid:18) a b c d. : a, b, c, d r} (cid:19) (cid:18) a + a(cid:48) (cid:18) aa(cid:48) + bc(cid:48) ab(cid:48) + bd(cid:48) (cid:19) + b(cid:48) d + d(cid:48) c + c(cid:48) ca(cid:48) + dc(cid:48) cb(cid:48) + dd(cid:48) and c d. [x, y ] = xy y x. Prove that (a) for all x, y m, we have [x, y ] = [y, x] and in particular, the operation is not commutative. (b) there exist elements x, y, z m such that. [[x, y ], z] (cid:54)= [x, [y, z]] and so the operation is not associative. (c) prove that for all x, y, z m, we have the identity.

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