Given a set x, show that the set of all surjectives: x→x form a monoid

Question

I am pretty stumped on this question. All I knew was the definition of a monoid consisted of 3 properties:

1. a Op b = c | Performing some operation on 2 values = some value c
2. Zero = neutral element, where Zero Op a = a
3. (a Op b) Op c = a Op (b Op c) | associativity

How do these properties help me show that a set of all surjectives x->x form a monoid? I was told there was composition involved, but I just have no clue how to go about it.

Answer 1

Guide:

The binary operation that you are looking for is the composition function.

You have to check the following:

• If $f$ is a surjection and $g$ is a surjection, can you prove that $f \circ g$ is a sujrection?

• Show that the identity map, $e$, is a surjection. Also, $e \circ f = f \circ e$.

• Check that $$(f \circ g) \circ h = f \circ (g \circ h)$$