Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$ (the complement of $X$)?
Would the inverse of the function just be the function itself?
Asked
Active
Viewed 34 times
0
-
Formatting tips here. – Em. Apr 27 '16 at 01:23
-
Please put more details in the body of the question. By that I mean, include the title in your body. – KKZiomek Apr 27 '16 at 01:25
-
I can't get the bar to go right... – Emily Apr 27 '16 at 01:39
-
When you say \barX do you mean $\bar X$? That usually denotes complex conjugate or else topological closure (in which case we need a topology). When you say P(Z) do you mean polynomials in one variable with integer coefficients, or the power set of $\Bbb{Z}$ (the set of subsets of $\Bbb Z$), or something else? – ForgotALot Apr 27 '16 at 01:48
-
Yes, I mean bar of X, or X compliment. I also mean the powerset of all integers, so what you said, yes. – Emily Apr 27 '16 at 01:57
1 Answers
1
The inverse function $\theta^{-1}$ is just the function $\theta$ itself, since the complement of the complement of a set $X$ is the set $X$ itself. For example, the complement of $X=$ the set of even integers is the set of odd integers. The complement of the latter set $\bar X$, i.e., the set of integers which are not odd, is the set of even integers $=X$.
ForgotALot
- 3,931