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I have a list of set identities that I need to apply to prove the left-hand side is equal to the right-hand side. I am stuck on which rules to use.

For all sets $A,B,C$, show that $$( A - B ) - C = A - ( B ∪ C ).$$

I found this document of set identities and I see that $A - B$ is equal to $A \cap B^C$, but then I get stuck. I also do not know if I am going in the right direction.

Daniel W. Farlow
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chasethesunnn
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3 Answers3

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$$x \in (A-B)-C \Leftrightarrow x \in A-B \wedge x \notin C \Leftrightarrow x \in A \wedge x \notin B \wedge x \notin C \Leftrightarrow x \in A-(B \cup C)$$

evinda
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Here is a very easy way to prove your identity using basic set algebra: \begin{align} (A-B)-C&\equiv (A\cap B^C)\cap C^C\tag{by definition}\\[0.5em] &\equiv A\cap(B^C\cap C^C)\tag{by assoc. of $\cap$}\\[0.5em] &\equiv A\cap(B\cup C)^C\tag{DeMorgan}\\[0.5em] &\equiv A-(B\cup C)\tag{by definition} \end{align} Did all of that make sense? Feel free to comment if a step was unclear.

Daniel W. Farlow
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  • This was the clearest solution, although I did understand the other answers. Thank you so much. – chasethesunnn Sep 09 '15 at 23:18
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    You're welcome! I'm glad you found it helpful--I find that writing notes in the margin is always a good idea to force you to make sure you know how each step works (such as how associativity comes into play, etc.). – Daniel W. Farlow Sep 09 '15 at 23:21
  • I think I will do that actually! I just tried the next problem set and it was super easy after using your answer as a guide on how to process the problem. :] – chasethesunnn Sep 09 '15 at 23:28
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    Excellent! I would recommend the practice of making margin notes/comments regardless of whether or not you're proving a set identity. If you are taking a discrete class, then you will no doubt come across induction. When you do induction proofs, actually writing the proof gets to be a little tricky. The same practice is very helpful (see this post for an example). As always, provide clarity whenever possible! Margin notes seem to do just that (i.e., provide clarity). – Daniel W. Farlow Sep 09 '15 at 23:31
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You have $(A - B) = A\cap B^{c}$. Apply this to the right hand side of the equality you need to prove to obtain $$ A-(B\cup C) = A\cap(B\cup C)^{c}. $$ By De Morgan's Law, $(B\cup C)^{c} = B^{c}\cap C^{c}$. Therefore $$ A - (B\cup C) = A\cap B^{c}\cap C^{c} = (A-B)-C. $$

Jesse
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