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What is the fundamental difference between a relation and function? Is every relation a 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,
(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,
(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.