inverse image question, is it correct?

**armeros** · June 20th 2009, 05:04 AM

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.

**Plato** · June 20th 2009, 05:55 AM

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.
$f_x(1,a)=f_x(1,b)=1~\&~f_x(2,a)=f_x(2,a)=2$ .

**armeros** · June 20th 2009, 07:47 AM

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!

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