# inverse image question, is it correct?

• June 20th 2009, 05:04 AM
armeros
inverse image question, is it correct?
Let $X$ and $Y$ be sets , let $A \subseteq X$ and $B$ $\subseteq Y$, and let $f:X \times Y \longrightarrow X$ be a projection map.

Show that $f^{-1}(A) = A \times Y$.

I wonder if the question is correct. For example, let $X$= {1, 2} and $Y$ = {a,b}. Then, $X \times Y$ = {(1,a), (1,b), (2,a), (2,b)}.

Suppose $f(1,a)= f(1,b)=f(2,a)=2,$ and $f(2,b)=1$.

Suppose $A$= {1} $\subseteq$ {1,2} = $X$.

Then $f^{-1}(A)$= {(2,b)} $\neq A \times Y$ = {(1,a), (1,b)}.

Please check whether I am right that the proposition was wrong from the question.
• June 20th 2009, 05:55 AM
Plato
Quote:

Originally Posted by armeros
Let $X$ and $Y$ be sets , let $A \subseteq X$ and $B$ $\subseteq Y$, and let $f:X \times Y \longrightarrow X$ be a projection map. Show that $f^{-1}(A) = A \times Y$.

I wonder if the question is correct. For example, let $X$= {1, 2} and $Y$ = {a,b}. Then, $X \times Y$ = {(1,a), (1,b), (2,a), (2,b)}.

You have the projection wrong.
$f_x(1,a)=f_x(1,b)=1~\&~f_x(2,a)=f_x(2,a)=2$.
• June 20th 2009, 07:47 AM
armeros
Oh, how careless I am.

I thought a projection is just another name for a function. The way I define in my inquiry was to define a function, not a projection.

Sorry to bother you with such a careless question.

Thanks a lot!