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reenmachine
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Homework Statement
Prove that if A and B are sets , then ##A \subseteq B \ \ \leftrightarrow \ \ A - B = \varnothing##
The Attempt at a Solution
Let ##A \subseteq B## be arbitrary.The definition of ##A \subseteq B## implies that ##\forall x \in A## , ##x \in B##.This implies that ##A - B = \varnothing## , which also disprove that ##A - B ≠ \varnothing##.Since we disproved the previous statement , we proved that ##A \subseteq B \ \ \leftrightarrow \ \ A - B = \varnothing##.
I think I understand the logic of why the statement is true pretty clearly , but I'm not sure if I took a shortcut somewhere or made a bad notation choice (or any other mistakes).Or do I have to explain why ##\forall x \in A## , ##x \in B## implies all of this?
Any thoughts would be appreciated.Thank you!
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