MHB Inverse of F: {(1,2)(2,2)(3,2)(4,5)(5,3)}

[JProgrammer]

F has the following sets:

F = {(1,3)(2,2)(3,2)(4,2)(5,5)}

Does F^-1 mean:

F = {(1,2)(2,2)(3,2)(4,5)(5,3)}

Thank you.

 

[Evgeny.Makarov]

F has the following sets:

F = {(1,3)(2,2)(3,2)(4,2)(5,5)}

Several remarks.

1. $F$ is probably a binary relation.
2. Elements of a set are listed in curly braces and are separated by commas.
3. Things like $(1,3)$ are not sets, but ordered pairs. There is also a set $\{1,3\}$, but $\{1,3\}=\{3,1\}$ as sets, while $(1,3)\ne(3,1)$ as ordered pairs.

Does F^-1 mean:

F = {(1,2)(2,2)(3,2)(4,5)(5,3)}

The definition of the inverse of a binary relation $F$ is as follows.
\[
F^{-1}=\{(y,x)\mid (x,y)\in F\}
\]
That is, you need to take every ordered pair in $F$ and switch the first and second elements in it.

 

[HallsofIvy]

The given F is a function from the set {1, 2, 3, 4, 5} to the set {2, 3, 5} (which can be interpreted as a subset of {1, 2, 3, 4, 5}). We can think of it as changing 1 to 3 (or "changing 1 to 3" or "mapping 1 to 3"), 2 to 2, 3 to 2, 4 to 2, and 5 to 5. Its inverse, F^{-1}, usually read as "F inverse", goes the opposite way, changing 3 to 1, 2 to 2, 2 to 3, 2 to 4, and 5 to 5. It can be written as F^{-1}= {(3, 1), (2, 2), (2, 3), (2, 4), (5, 5)}.

That F^{-1} is not a function- it is, rather, the more general "relation" (every function is a relation, not every relation is a function). The difference is that, for a function, which can always be written as "y= f(x)", the same value of x cannot give different values of y: we cannot have 2= f(2) and 3= f(2).