## Related questions with answers

Let X={a, b, c}. Define a function S from $\mathcal{P}$(X) to the set of bit strings of length 3 as follows. Let Y$\subseteq$X. If a$\in$Y, set $s_1=0$; If a$\notin$Y, set $s_1=1$; If b$\in$Y, set $s_2=0$; If b$\notin$Y, set $s_2=1$; If c$\in$Y, set $s_3=0$; If c$\in$Y, set $s_3=1$. Define S(Y)=$s_1s_2s_3$.

Prove that S is onto.

Solution

VerifiedDEFINITIONS

A relation $f$ is a $\textbf{function}$ when $(a,b)\in f$ and $(a,c)\in f$ implies $b=c$.

The $\textbf{domain}$ is the set of all possible first elements in the ordered pairs.

Notation: dom $f$

The $\textbf{image}$ is the set of all possible second elements in the ordered pairs.

Notation: im $f$

The function $f$ is $\textbf{one-to-one}$ whenever $(x,b),(y,b)\in f$ implies $x=y$.

The function $f:A\rightarrow B$ is $\textbf{onto}$ if for element $b\in B$ there exist an element $a\in A$ such that $f(a)=b$.

The function $f$ has an $\textbf{inverse function}$ if and only if $f$ is one-to-one and onto.

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