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Hypotheses: $A$ ⊆ $C$, $B$ ⊆ $C$, $A$ ∪ $B$ = $C$ and $A$ ∩ $B$ = ∅

Proof:

  1. Let $x ∈ A ⇒ x ∈ C ⇔ x ∈ A ∪ B.$

Since $x ∈ A ∧ x ∈ A ∪ B ∧ x ∉ A ∩ B ⇒ x ∉ B.$

Since $x ∈ C ∧ x ∉ B ⇒ x ∈ C - B.$

  1. Let $x ∈ C - B ⇔ x ∈ C ∧ x ∉ B ⇒ x ∈ C ⇔ x ∈ A ∪ B.$

Since $x ∈ A ∪ B ∧ x ∉ B ⇒ x ∈ A.$

Therefore $A = C - B.$

Is this proof right? if yes, is it abbreviated? should I put more details in it?

Undergraduate Student
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    That's all correct, I think it could be written up a bit more cleanly but it's all there – Stephen Donovan Jun 21 '21 at 21:04
  • @StephenDonovan Thanks! everyone tell me there is a problem in my writing, but I can't tell, or will it be fixed by time with more practicing? – Undergraduate Student Jun 21 '21 at 21:09
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    Typically my suggestion would be to write things out in full sentences more and inject more English instead of symbolic arguments, but in this specific case since the argument is a lot more akin to a propositional logic proof it might make the most sense to write it up like one of those, and give each step in the argument its own line, number, and rule. Do you know what kind of format I'm referring to? – Stephen Donovan Jun 21 '21 at 21:15
  • @StephenDonovan I think you mean the "Deductive Reasoning Method", but I try to go over it (reference rules are known implicitly) so I could be like real Mathematicians who use more English in proofs. Maybe I'm not doing it right. – Undergraduate Student Jun 21 '21 at 21:24

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