ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

We consider the unary operation − on R and want to determine those mappings
f of R into it self, which are homomorphisms. the condition is that
f(−x) = −f(x),
which is the condition for f to be an odd function
This result can be generalized to general linear spaces.
Exercise 9: Homomorphisms for unary operations (see nr 8)
Write the condition f : R → R to be a homomorphism wrt the operations
x ↦→ −x and x ↦→ x −1 , and find such a homomorphism. (Hint: f(x) = a x )
Exercise 10: The four fundamental isomorphisms concerning the algebraic operations.
(see nr 8)
Let a be a real constant. Study the four function definitions ax, a x , log a(x), x a
and explain that they define homomorphisms for suitable choices of domain
and codomain. Notice that all combinations of addition and multiplication are
possible.
Exercise 11: Closed wrt to addition. (see nr 14)
Construct an increasing chain of subsets of C which are closed wrt addition in
C
Exercise 12: Show that the set {x+y √ 3 : x, y ∈ Q} is closed both wrt addition
and multiplication .
Show that the same is not true for {x + y 3√ 2 : x, y ∈ Q}. (see nr 14)
Exercise 13: Product of equal factors (see nr 19)
Show that Z2 × Z2 = {(0, 0), (0, 1), (1, 0), (1, 1)}. Show that the product operation
+ × + partially is given in the following table and complete:
(+,+) (0,0) (0,1) (1,0) (1,1)
(0,0) (0,0) (0,1) (1,0) (1,1)
(0,1) (0,1) (0,0) (1,1) (1,0)
(1,0) (1,0) (1,1)
(1,1) (1,1) (0,1)
(·, ·) (0,0) (0,1) (1,0) (1,1)
(0,0) (0,0) (0,0) (0,0) (0,0)
(0,1) (0,0) (0,1) (0,0) (0,1)
(1,0) (0,0) (0,0)
(1,1) (0,0) (1,0)
Exercise 14: Product of different factors (see nr 19)
Make tables for (Z2, +) × (Z3, +) and (Z2, ·) × (Z3, ·), in the same way as in
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Example 15: Chinese remainder theorem (see nr 21)
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