How to show this mathmatically?

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To show that A \ B = ∅ if and only if A ⊆ B, one must demonstrate that if A is a subset of B, then removing elements of A that are also in B results in an empty set. The reasoning is that if every element of A is contained in B, then no elements remain when A is subtracted from B. Conversely, if A \ B is empty, it indicates that there are no elements in A that are not also in B, confirming that A is indeed a subset of B. The mathematical expression can begin with the definition of subset and proceed to show the implications clearly. This establishes the equivalence between the two statements.
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Homework Statement



Let A and B be sets.
Show that A \ B = ∅ if and only if A ⊆ B.


I think this means that A is a subset of B, therefor if I remove all the elements of A that are in B, A would end up being empty (or ∅).

How do I write this mathmatically or is the above sentence acceptable?
 
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Kaldanis said:

Homework Statement



Let A and B be sets.
Show that A \ B = ∅ if and only if A ⊆ B.I think this means that A is a subset of B, therefor if I remove all the elements of A that are in B, A would end up being empty (or ∅).

How do I write this mathmatically or is the above sentence acceptable?

You would start with something like, A is a subset of B implies for all x in A, x is in B. ...
 

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