Jesus wrote:

> Another for **Puzzle 141**, in **Set** for a surjection defining a labeled partition, the function in the converse direction sending partition labels to any of the elements in its block is a left inverse.

I may be getting confused, since I tend to mix up left and right, but the function in the converse direction seems to be a _right_ inverse! If so, you've solved Puzzle 142, not Puzzle 141. But it's nice either way.

To pick a very specific example, let's use this surjection:

\[ f: \\{a,b\\} \to \\{c\\} . \]

Any function

\[ g: \\{c \\} \to \\{a,b\\} \]

is a right inverse of \\(f\\), meaning that

\\[ f \circ g = 1_{\\{c\\}} . \\]

To see this, note that

\[ (f \circ g)(c) = c \]

no matter what \\(g\\) is.

> Another for **Puzzle 141**, in **Set** for a surjection defining a labeled partition, the function in the converse direction sending partition labels to any of the elements in its block is a left inverse.

I may be getting confused, since I tend to mix up left and right, but the function in the converse direction seems to be a _right_ inverse! If so, you've solved Puzzle 142, not Puzzle 141. But it's nice either way.

To pick a very specific example, let's use this surjection:

\[ f: \\{a,b\\} \to \\{c\\} . \]

Any function

\[ g: \\{c \\} \to \\{a,b\\} \]

is a right inverse of \\(f\\), meaning that

\\[ f \circ g = 1_{\\{c\\}} . \\]

To see this, note that

\[ (f \circ g)(c) = c \]

no matter what \\(g\\) is.