# Binary Structures and Mapping

• Sep 21st 2011, 06:17 PM
tangibleLime
Binary Structures and Mapping
I am having great difficulty conceptualizing binary structures and mapping from one to the other.

Here's my problem:

The map $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $\phi(n) = n + 1$ for $n \in \mathbb{Z}$ is one-to-one and onto $\mathbb{Z}$. Give the definition of a binary operation $*$ on $\mathbb{Z}$ such that $\phi$ is an isomorphism mapping.

$<\mathbb{Z},\cdot> with <\mathbb{Z},*>$.

So I have $<\mathbb{Z},\cdot>$. This means that for members a and b, this structure is $a \cdot b$ and $d \cdot a$, right? And I'm trying to map this to an unknown structure? I don't understand what exactly I'm doing.

What is that $\phi(n) = n + 1$ function? I assume it takes a number in $\mathbb{Z}$ and maps it to a number in $\mathbb{Z}$, specifically one more than the input number. But what does that have to do with my structures? What am I trying to find?

In utter confusion, I took $<\mathbb{Z},\cdot>$ and figured that for the inputs of a=2 and b=3, the answer would be 6 since 2*3=6. So I take that 6... add one to it from $\phi$, and that gives me 7... is that 7 the value that is supposed to correspond with the second structure, $<\mathbb{Z},*>$? So is my task to find a function $*$ that takes two variables $a$ and $b$ to make 7, or more abstractly, ab+1?? (Headbang)

Any help = appreciated.
• Sep 21st 2011, 10:43 PM
Deveno
Re: Binary Structures and Mapping
what you are being asked to do is FIND a definition for * so that φ(ab) = φ(a)*φ(b).

the fact that φ is 1-1 and onto already is the "iso" part....ensuring that φ(ab) = φ(a)*φ(b) is the "morphism" part.

suppose that a*b is defined as ab+1...does this definition of * fit the requirements?

(your question is a little vague...since no other information is given, i am assuming you aren't requiring that * be associative, or have an identity).