# Prove that there exists a minimum cut (S', T') where S' is the subset of S in which (S, T) is a minimum cut as well

I need to prove the statement $$\exists$$ a minimum cut $$(S', T')$$ where $$S' \subseteq S$$ for any minimum cut $$(S, T)$$

My attempt: Before proving this statement, I have a lemma if $$(S_1, T_1)$$ and $$(S_2, T_2)$$ are both minimum cuts, then $$(S_1 \cap S_2, T_1 \cup T_2)$$ and $$(S_1 \cup S_2, T_1 \cap T_2)$$ are minimum cuts as well.

My proof: let $$(S_1, T_1)$$ and $$(S_2, T_2)$$ be two minimum cuts. Based on the lemma, we have $$(S_1 \cap S_2, T_1 \cup T_2)$$ is a minimum cut. Also, $$S_1 \cap S_2 \subseteq S$$. Thus, the statement is correct. Is my proof correct?

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• Can't you just choose $(S',T') = (S,T)$? – Yuval Filmus Oct 13 '20 at 6:57