Prove/Disprove this function is surjective

Question

I'm having difficulty with figuring out an approach to solving this problem from my textbook. I understand what makes a function surjective, but that's understanding what the range is. Most examples I see proving or disproving "surjectivity" have more specific domains/codomains than just "sets".

Let X be a set. Define a map X → X × X by d(x) = (x, x).

Is d(x) surjective?

Answer 1

Let $X$ be any set.

Let $(a,b) \in X \times X$. (i.e. $a \in X, b\in X$).

Question is: Is there/must there be a $c \in X$ so that $d(c) = (a,b)$.

Well, $d(c) = (c,c)$ and so if $(c,c) = (a,b)$ then $c =a$ and $c =b$.

So for every $a,b\in X$ does there exist a $c \in X$ so that $c=a$ and $c = b$?

If $a \ne b$ then .... no.

So $X$ has more than $1$ element then $d$ is not surjective.

But if $X = \{x\}$ then $X \times X = \{(x,x)\}$ and $d = \{(x,(x,x))\}\subset X\times(X\times X)$ is surjective.

Answer 2

If $X$ has more than one element, say $x$ and $y$ are two different elements of $X$, then $(x,y)$ is not in the range, so the answer is no.

If $X$ has at most one element, then the answer is rather trivially yes.