Tagged Questions
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
2
votes
3answers
82 views
The inverse of the inverse in a group
In one of my math exercises, I'm being asked to prove that
for all $a, b \in G | (a^{-1})^{-1} = a$
with G a group. However, nowhere is stated that it is a commutative group. My first thought was ...
4
votes
2answers
168 views
Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?
I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
1
vote
1answer
115 views
Terminology question; inverse vs complement in Boolean algebra
This was said at a lecture I attended:
$e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$.
So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
2
votes
3answers
118 views
Determine invertible and inverses in $(\mathbb Z_8, \ast)$
Let $\ast$ be defined in $\mathbb Z_8$ as follows:
$$\begin{aligned} a \ast b = a +b+2ab\end{aligned}$$
Determine all the invertible elements in $(\mathbb Z_8, \ast)$ and determine, if possibile, ...
2
votes
2answers
80 views
15Puzzle, sum of inversions - what's been summed?
According to this page: http://mathworld.wolfram.com/15Puzzle.html it says that
While odd permutations of the puzzle are impossible to solve
I've red this article: ...
