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**ziabing****Member**- Registered: 2021-09-18
- Posts: 6

Hello!

I'm trying to prove that the following function is bijective and find its inverse function.

Let X be a set and A a subset of X.

f : P(X) → P(X)

A → the complement of set A

It's the first time I'm asked to do this with power sets and sets, so I have no idea of what I'm supposed to do.

All help is much appreciated!

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,353

hi ziabing

Sorry this didn't get answered ages ago. I found the post again when looking at your recent post.

A power set is a set of sets. It's members are all the subsets (including the empty set and the whole set) .

For every subset there is a complement set.

eg. If X = {w,x,y,z} and if A = {x,y} is a subset and A' = {w,z} is it's complement.

The mapping maps every such subset to its complement. So if you take any member of the complement set (A') let's say {x,y,z} ,there exists a member of A that maps onto it; in this example {w}

So the function is surjective.

Similarly if A = {w} is a set in X then it maps onto {x,y,z} in the complement set so it is injective.

The inverse function is the same function again as (A')' = A

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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