What is the fundamental difference between a relation and function?

Fundamental difference between Relation and Function:

Every function is a relation, but every relation need not be a function.

A relation f from A to B is called a function if

(i) Dom(f) = A

(ii) no two different ordered pairs in f have the same first component.

For. e.g

Let A = {a, b, c, d} and B = {1, 2, 3, 4, 5}

Some relations f, g and h are defined as follows

f = {(a, 1), (b, 2), (c, 3), (d, 4)}

$g=\{(a, 1),(b, 3),(c, 5)\}$

$h=\{(a, 1),(b, 2),(b, 3),(c, 4),(d, 5)\}$

In the relation f,

$f=\{\underline{(a, 1),}(\underline{b}, 2),(c, 3),(d, 4)\}$

(i) Dom (f) = A

(ii) All first components are different. 

So, f is a function

In the relation g,

(i) Dom (g) ≠ A

So, the condition is not satisfied. Thus, g is not a function.

In the relation h

$h=\{(a, 1),(b, 2),(b, 3),(c, 4),(d, 5)\}$

(i) Dom (h) = A

(i) Two first components are the same, i.e. b has two different images.

So, h is not a function.

No, every relation is not a function.