IB Maths HL Option Sets Isomorphism

Operation table of a group, Symmetries of plane figures, Cycle notation for permutations, Left and right cosets, group homomorphism, kernel, kernel and range of a homomorphism, isomorphisms

IB Maths HL Option Sets Isomorphism

IB Mathematics HL Option: Sets, Relations and groups (Groups-Isomorphism)

How can we show that the function $f:\mathbb{R} -> \mathbb{R}^{+}$ defined by $f(x)=e^x$ is an isomorphism between $\{ \mathbb{R} , + \}$ and $\{ \mathbb{R}^{+} ,* \}$

Thanks
elizabeth

Posts: 0
Joined: Mon Jan 28, 2013 8:09 pm

Re: IB Maths HL Option Sets Isomorphism

IB Mathematics HL Option: Sets, Relations and groups (Groups-Isomorphism)

The definition of isomorphism is
Two groups {G,*} and {H,o} are isomorphic if:
- there is a bijection $f:G -> H$
and $f(x*y)=f(x) o f(y)$ for every $x,y \in G$

and for your question we have
$\{ \mathbb{R} , + \}$ is a group under addition
and $\{ \mathbb{R}^{+} ,* \}$ is a group under multiplication, and $f$ is a bijection from $\mathbb{R}$ into $\mathbb{R}^{+}$

$f:\mathbb{R} -> \mathbb{R}^{+}$ defined by $f(x)=e^x$ is an

isomorphism between $\{ \mathbb{R} , + \}$ and $\{ \mathbb{R}^{+} ,* \}$

Since f has inverse function $f^{-1} (x)=lnx$ and we have the following

$f (x+y)=e^{x+y}=e^x \cdot e^y =f(x) \cdot f(y)$

Hope these help
miranda

Posts: 268
Joined: Mon Jan 28, 2013 8:03 pm