I have a couple of questions on elementary set theory.
Q1: Let $f: X \to Y$ be a function, and let $A$ be a subset of $X$, $B$ is a subset of $Y$. Find examples which show that the following statements are false.
a. If $B$ is not equal to an empty set, then $f^{-1}(B)$ is not equal to an empty set.
b. $f^{-1}(f(A))=A$
c. $f(f^{-1}(B))=B$
d. $f(X)=Y$
i am having a problem from a.
A solution says, if $f(x)=1$, $B=\{2\}$, $f(B)=1$. For all $x$, $f^{-1}(B)=\emptyset$.
why $f^{-1}(B)$ is an empty set?
Q2:
Find $f^{-1}(\{0,1\})$, $f^{-1}(\{-3,3\})$, and $f^{-1}(\{1,2,3\})$ from below function (venn diagram).
To descrbie the venn diagram in words..
(domain) Set $X={x,y,z,v,u}$
(range) Set $Y={a,b,c,d,e}$
$\{x\} \to \{d\} \\
\{y\} \to \{e\} \\
\{z\} \to \{a\} \\
\{u,v\} \to \{b\}$
I am having trouble finding the original elements.
this is what I found but i am not sure if they are right:
$f^{-1}(\{0,1\})=\{\{\pm \sqrt{2}\}, \{\pm \sqrt{3}\}\}\\
f^{-1}(\{-3,3\})=\{\{\pm \sqrt{5}\}\}\\
f^{-1}(\{1,2,3\})=\{\{\pm \sqrt{3}\}, \{\pm 2\}, \{\pm \sqrt{5}\}\}$
Q3: Let $X=\{x,y,z\}$, and $Y=\{1,2,3\}$. Which of the following constitute functions from $X$ to $Y$? If they do not, give the reason.
d) $i=\{(x,1),(x,2),(y,1),(z,3)\}$
It is not a function from $X$ to $Y$ because there two elements in the domain that have different ranges (different y-values). (would it be correct to say this? Also, if the two elements $(x)$ had the same $y$ value, let's say 1, –> $(x,1) \ \text{&} \ (x,1)$,
can i say that this is a function from X to Y?
Thank you very much for your insights!
Best Answer
To just do the first:
Let $X=\{a,b,c\}$, $Y=\{1,2,3\}$, and let the map be $a \to 1, b\to 1, c \to 2$.
Then $f[X]=\{\{1,2\} \neq Y$.
For $A=\{a\}$, $f[A]=\{1\}$ but $f^{-1}[f[A]]= f^{-1}[\{1\}]= \{a,b\} \neq A$.
For $B=\{3\}$ we have that $f^{-1}[B]=\emptyset$
For that same $B$ we also have that $f[f^{-1}[B]]=\emptyset \neq B$ as well. Also for $B=Y$ we have that $f[f^{-1}[B]]=f[X]\neq B$.
Hope this helps.