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Due: September 19, 2012
1. Consider the set
:a, b, c, d ∈R}
with the usual operations of addition and multiplication:
Deﬁne the new operation
[X, Y ] = XY −Y X.
(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
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