Identity Relation and Function

[FourierX]

Homework Statement

I am reading a book on relations on function and I am very confused with identity relation and function. Any help on understanding I relation and I function will be appreciated.

Homework Equations

A function from A to B is a relation f from A to B such that
a) the domain of F is A
b) if (x,y) $$\in$$ to F and (x,z) $$\in$$ f, then y = z

The Attempt at a Solution

x $$\in$$ A , I_A(x)= x

 

[lurflurf]

I am very confused by your question. What are f and F?
A relation f from A to B is a subset of AxB
A function f is a relation from A to B such that
a)The domain of f is A.
b)If (x,y) and (x,z) are elementa of f, then y=z.

So a relation is a special type of function. All functions are relations, but not all relations are functions.

Say we have a relation R from A to B.
R is a subset of AxB.
This means if we chose x in A and y in B it makes sense to ask questions like
Is (x,y) in R?
Is (x,z) in R?
Is x in the domain of R?
Any combination of answers is possible, but say we have a relation f from A to B.
and ask
1)Is (x,y) in R?
2)Is (x,z) in R?
3)Is (x,y)=(x,z)?
4)Is x (x is in A) in the domain of R?
4 is always true
If at least 2 of 1,2,3 are true then they all are

Say we have a relation marrage from men to women
we can have
(Bob,Jill) and (Bob,Beth) in marrage

We might have Bob is not married

Say we have a function marrage from men to women
we cannot have
(Bob,Jill) and (Bob,Beth) in marrage unless Jill=Beth
but we can have
(Bob,Jill) and (Sam,Jill) with Bob!=Sam
We cannot have Bob is not married.
We can have Jill is not married.

 

[HallsofIvy]

In other words, don't use "f" and "F" to mean the same thing!

Perhaps what is confusing you is that a "function" is a special type of "relation".

Every function is a relation but the other way is not true: the relation {(x,y)|x²+ y²= 1} is not a function.

The identity function {(x,x)} for all x in the base set is also the identity relation- there is no difference.