How to Reverse a Proof for an Identity with Sets?

[ainster31]

Homework Statement

Homework Equations

The Attempt at a Solution

$$(A-B)\cup (C-B)=(A\cup C)-B\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\$$

I know for algebraic proofs, proofs like these are accepted if they are reversed. But how would I reverse this proof?

 

[tiny-tim]

hi ainster31!

I know for algebraic proofs, proofs like these are accepted if they are reversed. But how would I reverse this proof?

let's reverse it, as you say, and then analyse it …

$$ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\(A-B)\cup (C-B)=(A\cup C)-B$$
from the first line to the second is the distributive rule

from the second to the third is simply applying the definition of "minus"

(but the best way would be to start from bottom left, go up the left side, and cme down the right side (or vice versa))​

 

[ainster31]

hi ainster31!


let's reverse it, as you say, and then analyse it …

$$ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\(A-B)\cup (C-B)=(A\cup C)-B$$
from the first line to the second is the distributive rule

from the second to the third is simply applying the definition of "minus"

(but the best way would be to start from bottom left, go up the left side, and cme down the right side (or vice versa))​

That makes perfect sense. Thanks!