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September 24th, 2017, 01:24 AM   #1
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Injective relation

Hello, I'm trying to work on the following:

Let A be a subset of a set E. F is the relation that associates each X of P(E) with the symmetric difference of A and X. Show that F is injective.

My idea has been to use the definition of an injective relation to show that for X and X' in P(E), if F(X) = F(X') then X = X'. Knowing that these two sets are equal if and only if their indicator functions are equal, I found a general formula for the indicator function of a symmetric difference, so that I can use it here. But I'm stuck.
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September 24th, 2017, 10:15 AM   #2
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Quote:
Originally Posted by Mifarni14 View Post
My idea has been to use the definition of an injective relation to show that for X and X' in P(E), if F(X) = F(X') then X = X'.
Right, that's the definition of injective so that's the way to get started.


Quote:
Originally Posted by Mifarni14 View Post
Knowing that these two sets are equal if and only if their indicator functions are equal, I found a general formula for the indicator function of a symmetric difference, so that I can use it here.
That sounds overly complicated. Does your formula work in the infinite case?

It's not hard to show that $A \bigtriangleup X$ and $A \bigtriangleup X'$ are different if $X \neq X'$. That's the way to go here.

Last edited by Maschke; September 24th, 2017 at 10:18 AM.
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