For all values of 'a' of element x , there exists a function g such that g: x -> x such that g(a) = a. Is this true?

Question

Let $X$ be a non-empty set. $ \forall a \in X, \exists \, g : X \rightarrow X$ such that $g(a) = a$.

Is this statement:

• a) always true
• b) always false
• c) the answers depend on the number of elements in $X$?

Answer 1

Always true.

For a trivial example, take

$g(x) = a, \; \forall a \in X; \tag 1$

for a less trivial example, take

$g(x) = x, \; \forall x \in X, \tag 2$

then

$g(a) = a; \tag 3$

if $X$ has at least two distinct elements $a$ and $b$, set

$g(x) = x, \; \forall x \in X \setminus \{a, b \}, \tag 4$

set

$g(a) = b, \tag 5$

and

$g(b) = a; \tag 6$

this provides an example which does not fix $a$.

As long as $X \ne \emptyset$, the existence of $g:X \to X$ with $g(a) = a$ for any $a \in X$ does not depend on the number of element of $X$.

Answer 2

The function $g$ is just identity. So we can define identity for any non-empty set.
Example: Let $X=\mathbb{R}$.
Then for $x\in \mathbb{R}$, we can write $g(x)=1_{\mathbb{R}}(x)=x$