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I have a terrible time converting my thought process into a proof. I can see how this statement is true, but writing out an actual proof I get lost pretty easily. This is what I have:

Assume that $A \subseteq B$ then $A \cap B = A$. We will show that $A \cap B \subseteq A$ and $A \subseteq A \cap B$. Assume that if $x \in A$ and $x \in B$ then $x \in A \cap B$ is a subset of A. For $A \subseteq A \cap B$ we assume $x \in A$ then $x \in A \cap B$. Since $A \subseteq B, x \in B$ then $x \in A \cap B$.

Assume $A \cap B = A$ we must show that $A \subseteq B$. Let $x \in A$ then $x \in A \cap B$ since $A = A \cap B$ then $x \in B$.

Am I on the right track to having this be an actual proof? If not where did I screw up and how do I make it make sense?

Michael Rector
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  • Seems fine to me, I would simply not introduce $x$ when demonstrating $A\cap B\subset A$ – Astyx Oct 29 '16 at 17:39

1 Answers1

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A couple comments:

  • You say "Assume $A \subseteq B$ then $A \cap B = A$." You only want to assume $A \subseteq B$, then show $A \cap B = A$.
  • When you show $A \cap B \subseteq A$, your wording could be better. You say "Assume that if $x \in A$ and $x \in B$ then $x \in A \cap B$ is a subset of $A$." It sounds like you are assuming what you want to show. Instead you should choose an arbitrary element $x \in A \cap B$. Then $x \in A$, which shows $A \cap B \subseteq A$.
  • This is not a mathematical comment, but a style comment. A lot of your sentences are run-ons or should be split into two sentences, e.g. "Assume $A \cap B=A$ we must show that $A \subseteq B$" or the sentence I quoted in the last bullet point. Don't be afraid of using short sentences in proofs. The goal is to make your logic as clear as possible to a reader.
kccu
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