1
$\begingroup$

For each of the following sets, determine if the given operation is a binary operation or not. Explain your answers.

(i) With A a non-empty set, the set of all functions $f : A \to A$ which are surjective (or onto), with composition.

(ii) With A a non-empty set, the set of all functions $f : A \to A$ which are injective (or one-to-one), with composition.

I'm kind of lost here out of the gate i belive we are talking about $F_A$ which i thought was the set of all functions that map $A \to A$

For i) i think we restrict $F_A$ to contain only functions that map from A onto all of A with the binary operation being composition of theses functions ( where we can think of each onto function as an element in $F_A$ ) since every function is onto every composition will be onto so he binary operation of composition should be well defined?

For ii) i think it is a binary operation cause each composition will always map all the values it is given back to the set A. ( Even if the composition doesn't yield all of A we should never get anything not in A out of the composition.)

  • 0
    You state that you're asking *determine if the given operation is a binary operation*. However, from your discussion, it sounds like you're interested in deciding whether the given set is **closed under** the given binary operation. Is this what you're really asking, or are you just interested in addressing the problem statement? – 2017-01-11
  • 0
    Ah Its a recommended problem from my class i'm more interested in deciding if the given set is closed under the given binary operation but im not even sure it is a binary operation yet? – 2017-01-11
  • 0
    It's not clear where you're having trouble with that part, though. Have you looked back at the definition of a "binary operation"? Are you having trouble seeing how this definition applies to composition? – 2017-01-11
  • 0
    I think im mostly confused by $F_A$ and what exactly this means in a given example? – 2017-01-11

1 Answers 1

1

Maybe writing some notation will help.

Let $f, g \in F_A = \{f \mid f: A \to A \text{ is surjective}\}$ and let $B : F_A \times F_A \to F_A$ be the binary operation where $B(f,g) = f \circ g$.

To determine closure, you need to determine if $f \circ g : A \to A$ is surjective, where $(f \circ g)(a) = f(g(a))$ for $a \in A$.