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Abstract Algebra

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One-to-One or Injective Function

ABSALG-MGOJVO

A function $f:S\rightarrow T$ such that $f(s)=f(s')$ implies $s=s'$ is called one-to-one (also written $1-1$).

One-to-one functions are also called injective functions, and they occur in Abstract Algebra, for example in the context of linear transformations of vector spaces or of homomorphisms of groups (don't worry if you haven't met such examples yet).

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

​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

$f:S\rightarrow T$, where $S=\{-\pi\le \theta\le \pi\}$, $T=\{t\in{\mathbb C}\mid |t|=1\}$, and $f(\theta)=e^{i\theta}$

D

$f:I\rightarrow T$, where $I=\{x\mid 0\le x< 1\}$, $T=\{t\in{\mathbb C}\mid |t|=1\}$, and $f(x)=e^{2\pi i(x+\sqrt{2})}$

E

$f:P\rightarrow T$, 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 $a\not\in A$.