What, if anything, is wrong with the following condensed proof? enter image description here
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2Possible (unanswered) duplicate of [this question](https://math.stackexchange.com/questions/4421927/falseproofinduction). – Galen Apr 07 '22 at 02:02

1Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, *etc.*; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult [this link](https://math.meta.stackexchange.com/q/9959) for further guidance. @DifferentialCovariance not a dup, different question but obviously from the same assignment. – Andrew L Apr 07 '22 at 02:08
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The proof assumes that both $x$ and $y$ are in both $S_1$ and $S_2$. If this is true for all $x$ and $y$ then $S_1=S_2$. Otherwise we can choose an $x$ that is in $S_1$ but not $S_2$ (or vice versa) and the induction will fail.
marty cohen
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The proof assumes that both x and y are in S1 and S2. If this is true for all x and y, then S1=S2. Otherwise, we can choose an x that is in S1 but not in S2 (or vice versa), and the induction will fail. $$S_1=S_2=......=S_{n}$$ ??? – Josue Furiano Apr 07 '22 at 02:36