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set notation

The following questions require a true/false statement as well as a supporting claim.

I'm having a hard time understanding set notation as well as their identities. This is my understanding (tentative answers) so far:

  1. False, $A\cup B$ intersects $C$ means everything in $A$ and $B$ also happens to be in $C$. Then $A$ union ($B$ Intersects $C$) means all elements that are in both $B$ and $C$ in a union with all elements of $A$. However, I cannot put that into a clearer explanation.

  2. True, $\setminus$ means set difference. So therefore the union of $A$ and $B$ $- B$ is going to be $A$ because set difference is all elements in $A$ and $B$ that are not in $B$.

  3. I'm not even sure how to get started on this. From my understanding this is an axiom.

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petrov
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1 Answers1

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Problem 1. You have the right idea, but to correctly disprove a statement, you need a specific counter-example. To illustrate your idea, we could take $A=\{a\}$, $B=\varnothing$, and $C=\{a\}$. Then $$ (A\cup B)\cap C=\{a\}\cap\{a\}=\{a\} $$ while $$ A\cup(B\cap C\}=\{a\}\cap\varnothing=\varnothing $$ This proves that $(A\cup B)\cap C$ is not necessarily the same as $A\cup(B\cap C\}$.

Problem 2. I don't think your reasoning is quite sound here. Can you write down an example similar to the one I've provided above to illustrate this claim? Hint: What happens if $A\cap B\neq\varnothing$?

Problem 3. This is definitely not an axiom (be very careful with what you call an axiom). Again, can you write down a specific example? Hint: Try taking $m=n=1$. Then this is the statement $A_1\subseteq A_1\cap A_2$. Does this make sense?

Brian Fitzpatrick
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