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I have the following problem: List the elements of the set $\overline{A\cap B}\cup C$, where $\overline{X}$ denotes the complement of an arbitrary set $X$ and $U$ denotes the universe under consideration. The considered sets are as follows:

  • $U = \{1,2,3,\ldots,10\}$
  • $A = \{1,4,7,10\}$
  • $B = \{1,2,3,4,5\}$
  • $C = \{2,4,6,8\}$

I believe I have the answer but not too sure. Here's what I came up with: $$ \overline{A\cap B}\cup C = \{2,3,5,6,7,8,10\}. $$

Is this answer correct and more so written correctly?

Thanks in advance.

Daniel W. Farlow
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natur3
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  • Check this out for typesetting: http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Daniel W. Farlow Jan 18 '15 at 03:25
  • Assuming $\overline{A}$ means "complement of $A$", are you trying to figure out what $\overline{(A\cap B) \cup C}$ is or what $\overline{A\cap B}\cup C$ is? – Daniel W. Farlow Jan 18 '15 at 03:27
  • @induktio thanks for the link! I'm trying to figure out what the 2nd one is. I'm still trying to learn how to type problems in here correctly and how to write the answer itself. – natur3 Jan 18 '15 at 03:33
  • @induktio read the link thanks!! this is a life saver! – natur3 Jan 18 '15 at 03:36
  • To see how my edit was typeset, you can click on the "edit" button--do not edit the post unless you need to, but you can see how I edited things to look better. – Daniel W. Farlow Jan 18 '15 at 03:48

2 Answers2

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Your goal is to determine what $\overline{A\cap B} \cup C$ is. The most sensible thing to do is approach it in a very piecemeal fashion:

  1. Determine what $A\cap B$ is.
  2. Determine what $\overline{A\cap B}$ is.
  3. Determine what $\overline{A\cap B} \cup C$ is.
  1. $A\cap B = \{1,4\}$
  2. $\overline{A\cap B} = U\setminus\{1,4\} = \{2,3,5,6,7,8,9,10\}$
  3. $\overline{A\cap B}\cup C = U\setminus\{1\} = \{2,3,4,5,6,7,8,9,10\}$

Does that all make sense?

Daniel W. Farlow
  • 22,531
  • Thanks again, yes this does except just to be clear does the evaluation of $\overline{A\cap B}$ check for union against C or the entire Universe set? I ask because I'm wondering why the 4 is in the final set but not the 1? I hope that makes sense. – natur3 Jan 18 '15 at 03:49
  • I'm not entirely sure what you're asking. In the interest of perhaps being somewhat clearer, consider this $$\overline{A\cap B}\cup C = {2,3,5,6,7,8,9,10}\cup {2,4,6,8} = {2,3,4,5,6,7,8,9,10}.$$ Does that make sense now? – Daniel W. Farlow Jan 18 '15 at 03:51
  • perfect! i just couldnt see it in my head; this cleared me up. thanks so much – natur3 Jan 18 '15 at 03:55
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With $U = \{1,2,3,...,10\}, A = \{1,4,7,10\}, B = \{1,2,3,4,5\}, C = \{2,4,6,8\}$

The question asks to find $(A\cap B)'\cup C$, where $'$ denotes complement.

We can go about this different ways. First way would be to do it step by step how it is currently written, following a sort of "order of operations" kind of feel.

$$A\cap B = \{1,4\}\\ (A\cap B)' = \{2,3,5,6,7,8,9,10\}\\ (A\cap B)'\cup C = \{2,3,4,5,6,7,8,9,10\}$$

Another good idea would be to algebraically modify the representation to make it look easier to work with.

$$(A\cap B)'\cup C = A'\cup B' \cup C\\ A' = \{2,3,5,6,8,9\}\\ B' = \{6,7,8,9,10\}\\ A'\cup B' \cup C = \{2,3,4,5,6,7,8,9,10\}$$

JMoravitz
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  • Thanks this is very helpful also. I would up vote this comment as well but not enough rep yet. – natur3 Jan 18 '15 at 03:38