Proving the preimage of a singleton under f is a singleton for all y

[cbarker1]

Homework Statement

Let #f:X \to Y# be a function. Show that if #f^{-1}({y})# is a singleton for all #y \in Y#.

Relevant Equations

Definition of Preimage is #f^{-1}(B)={ x\in X: f(x) \in B}# where B is a subset of Y.
#f^{-1}({y})={x}#

Dear Everyone,

I have some trouble how to start the proof of this statement. I need to prove the preimage of the singleton under f is the subset of singleton of x and vice versus. My attempt is this:Given y.

So we know that definition of the preimage is when all #x# is in #X# , then #f(x) \in B#.I am lost after these facts.

Thank for any assistance,

Cbarker1

 

[fresh_42]

Homework Statement:: Let #f:X \to Y# be a function. Show that if #f^{-1}({y})# is a singleton for all #y \in Y#.
Relevant Equations:: Definition of Preimage is #f^{-1}(B)={ x\in X: f(x) \in B}# where B is a subset of Y.
#f^{-1}({y})={x}#

Dear Everyone,

I have some trouble how to start the proof of this statement.

I'm not surprised. What is the full statement? Show if ... then?
What if I map the entire set $X$ onto a single point $y\in Y$?

I need to prove the preimage of the singleton under f is the subset of singleton of x and vice versus. My attempt is this:Given y.

So we know that definition of the preimage is when all #x# is in #X# , then #f(x) \in B#.I am lost after these facts.

Thank for any assistance,

Cbarker1

 

[cbarker1]

I'm not surprised. What is the full statement? Show if ... then?
What if I map the entire set $X$ onto a single point $y\in Y$?

The professor was typing quickly and forget then statement. If #f^{-1}({y})# is a singleton for all #y \in Y#, then #f# is bijection.

 

[PeroK]

The professor was typing quickly and forget then statement. If #f^{-1}({y})# is a singleton for all #y \in Y#, then #f# is bijection.

What two things do you have to show for $f$ to be a bijection?

 

[cbarker1]

Surjectivity and interjectivity

 

[fresh_42]

Surjectivity and interjectivity

Surjectivity and injectivity, yes. But what does that mean, what do you have to check?

 

[PeroK]

Surjectivity and interjectivity

Let's see you prove those.