mappings and relations

[erblina]

Let f: A→B be a map . Prove that f is injective if and only if ker f=ΔA , where ΔA={(x,x)|x∈A} and kerf is a relation defined with : x kerf y ⇔ f(x)=f(y)
Can someone tell me how to prove this ? I just know that the relation ker f is relation of equivalence that means it is reflexive , symetric and transitive .

 

[HallsofIvy]

It looks to me like it follows directly from the definition of "injective". What is that definition?

 

[erblina]

It looks to me like it follows directly from the definition of "injective". What is that definition?

for every x and y of A from : f(x)=f(y) follows x=y
then is f injective and i dont know if i can say that :
direct way :
if f is injective then
for every x and y from f(x)=f(y) follows x=y and from the definition of ker f
x ker f y (equivalent) with f(x)=f(y) from f injective follows x=y
so for every x and y :
(x,y)∈ker f follows x=y that means (x,x)∈ΔA
ker f⊆ΔA
is this direction true ? and how can i prove the other way that ΔA⊆ker f?

 

[HallsofIvy]

Let f: A→B be a map . Prove that f is injective if and only if ker f=ΔA , where ΔA={(x,x)|x∈A} and kerf is a relation defined with : x kerf y ⇔ f(x)=f(y)
Can someone tell me how to prove this ? I just know that the relation ker f is relation of equivalence that means it is reflexive , symetric and transitive .

Your definition of "kerf" only makes sense if A= B. Were you given that?