When is a relation a function?

Question

Can you explain the difference between a function and a relation and how a function is a subset of a relation, and when a relation is a function and when not?

Also, what is the domain, co-domain and range of a function?

All I’ve found on the internet and books has been a bunch of definitions which I can’t make sense out of. An intuitive explanation or any understandable illustration would be much appreciated.

Consider $${(1,2),(1,3)}$$as a relation not a function and $${(1,2),(3,4)}$$ as both a relation and a function. Can you point out the difference? — Mostafa Ayaz, Mar 23 '19 at 07:19
@Mostafa Ayaz There’s no number paired twice in the second one? — Brenda, Mar 23 '19 at 07:21
Not necessarily. Also ${(2,1),(3,1)}$ is a function. In fact if $(x,y)$ and $(a,b)$ are included in a function, then $x=a$ implies $y=b$. — Mostafa Ayaz, Mar 23 '19 at 07:29
@Mostafa Ayaz So for a set of ordered pairs to be a function, the first element of each can’t be paired with multiple elements? x = a would then imply y = b because it has to otherwise the set wouldn’t be a function? — Brenda, Mar 23 '19 at 07:39
That's right and followed by the definition of the functions... — Mostafa Ayaz, Mar 23 '19 at 07:44

score 0 · Answer 1 · answered Mar 24 '19 at 01:04

A relation R is a function when for all x,y,z, (xRy and xRz implies y = z).

The domain of a function f is the 1st projection of f.
{ x : exists y with (x,y) in f }

The range of a function f is the 2nd projection of f.
{ y : exists x with (x,y) in f }

The codomain of a function is arbitray as long as the range is a subset of it.

When is a relation a function?

1 Answers1