Identity Relation and Function

1. Dec 4, 2008

FourierX

1. The problem statement, all variables and given/known data

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.

2. Relevant 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

3. The attempt at a solution

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

2. Dec 4, 2008

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.
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.

3. Dec 4, 2008

HallsofIvy

Staff Emeritus
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.