1. ## Finding a Function

Let A= {a,b,c} and B= {1,2,3};$\displaystyle f : A \rightarrow B$ and $\displaystyle g: A \rightarrow B$ be a function defined by

$\displaystyle f$
$\displaystyle a \rightarrow 3$
$\displaystyle b \rightarrow 1$
$\displaystyle c \rightarrow 2$

$\displaystyle g$
$\displaystyle a \rightarrow 1$
$\displaystyle b \rightarrow 3$
$\displaystyle c \rightarrow 1$

a) Find the function $\displaystyle F: B \rightarrow A$ such that $\displaystyle F \circ f = 1_A$. What is $\displaystyle f \circ F$?
b) Find a function H: $\displaystyle A \rightarrow A$ such that $\displaystyle f \circ H = g$.

for (a) am I suppose to find out which one it mapps to cause if so I dont undertstand and I have a list of problems like this.

2. Originally Posted by tigergirl
Let A= {a,b,c} and B= {1,2,3};$\displaystyle f : A \rightarrow B$ and $\displaystyle g: A \rightarrow B$ be a function defined by

$\displaystyle f$
$\displaystyle a \rightarrow 3$
$\displaystyle b \rightarrow 1$
$\displaystyle c \rightarrow 2$

$\displaystyle g$
$\displaystyle a \rightarrow 1$
$\displaystyle b \rightarrow 3$
$\displaystyle c \rightarrow 1$

a) Find the function $\displaystyle F: B \rightarrow A$ such that $\displaystyle F \circ f = 1_A$. What is $\displaystyle f \circ F$?
b) Find a function H: $\displaystyle A \rightarrow A$ such that $\displaystyle f \circ H = g$.

for (a) am I suppose to find out which one it mapps to cause if so I dont undertstand and I have a list of problems like this.
I would prefer drawing a picture - but let's try writing symbols instead.
Try to think about what exactly $\displaystyle F \circ f = 1_A$ means
You can rewrite it as the following 3 conditions:
a=F(f(a))=F(3)
b=F(f(b))=F(1)
c=F(f(c))=F(2).
Now we know the values of F(1),F(2),F(3) - the function F is fully determined, isn't it?

Note that (if you draw a picture) F is the same as f, but with all arrows reversed.

I think you could be able to do the second part in a similar way.