The discussion focuses on reversing a proof for the identity involving sets, specifically the equation $$(A-B)\cup (C-B)=(A\cup C)-B$$. Participants analyze the proof using the distributive rule and the definition of set difference. The consensus is that the proof can be effectively reversed by starting from the bottom left and moving up the left side before coming down the right side. This approach clarifies the logical steps involved in the proof reversal process.
PREREQUISITESStudents of mathematics, particularly those studying set theory, algebra, and proof techniques, will benefit from this discussion. It is also valuable for educators looking to enhance their teaching methods in these areas.
ainster31 said:
tiny-tim said: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))