?

Abstract Algebra

Free Version

Upgrade subject to access all content

Easy

One-to-one Correspondence, Bijection

ABSALG-JE4SEK

A function $f:S\rightarrow T$ that is both one-to-one and onto is called a one-to-one correspondence (or bijective function or bijection). For some reason onto got dropped from the terminology one-to-one correspondence, so you should remember that onto is part of it nonetheless. You can think of a one-to-one correspondence as being a function $f$ matches every element $t\in T$ to just one $s\in S$.

Which of the following functions are NOT one-to-one correspondences?

​By ${\mathbb C}$ we mean the complex numbers, and $i=\sqrt{-1}$.

Select ALL that apply.

A

$f:S\rightarrow S$, where $S=\{1,i,-1,-i\}$ and $f(s)=s^4$

B

$f:S\rightarrow S$, where $S=\{1,i,-1,-i\}$ and $f(s)=s^3$

C

$f:P\rightarrow T$, where $P$ is the set of subsets of $U=\{1,2\}$, $T$ is the set of functions from $U$ to $\{0,1\}$, and $f$ associates to each $A\in P$ the function $\chi_A$, where $\chi_A(a)=1$, if $a\in A$ and $\chi_A(a)=0$, if $a\not\in A$.

D

$f:[0,1)\rightarrow T\times T$, where $T=\{t\in{\mathbb C}\mid |t|=1\}$, and $f(x)=(e^{2\pi ix}, e^{-2\pi i x})$.

E

$f:[0,1)\times [0,1)\rightarrow T$ where $T=\{t\in{\mathbb C}\mid |t|=1\}$, and $f(x,y)=e^{2\pi ixy}$.