 # Let A={0,2,3}, B={2,3}, C={1,5,9}, and let the universal set be U={0,1,2,...,9}. Greyson Landry 2022-07-17 Answered
Let $A=\left\{0,2,3\right\},B=\left\{2,3\right\},C=\left\{1,5,9\right\}$, and let the universal set be $U=\left\{0,1,2,...,9\right\}$. Determine:
$a\right)A\cap B\phantom{\rule{0ex}{0ex}}b\right)A\cup B\phantom{\rule{0ex}{0ex}}c\right)B\cup A\phantom{\rule{0ex}{0ex}}d\right)A\cup C\phantom{\rule{0ex}{0ex}}e\right)A-B\phantom{\rule{0ex}{0ex}}f\right)B-A\phantom{\rule{0ex}{0ex}}g\right){A}^{c}\phantom{\rule{0ex}{0ex}}h\right){C}^{c}\phantom{\rule{0ex}{0ex}}i\right)A\cap C\phantom{\rule{0ex}{0ex}}j\right)A\oplus B$
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Step 1
The Universal Set is the set that contains everything. So just imagine we are given a a universal set S; therefore all we know to exists is in the set S. That being said, any set A $\subseteq$ S we know that ${A}^{C}\cup A=U$ (this should be clear, but if it is not try to prove it). Now that we have this rule it should be clear that if:
$U=\left\{0,1,2,...,9\right\},A=\left\{0,2,3\right\}$
Step 2
Then ${A}^{C}=U,A=\left\{1,4,5,6,7,8,9\right\}$

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Step 1
${A}^{C}=U-A=$ Elements which are in set U but not in set A.
Step 2
Hence, ${A}^{C}=\left\{1,4,5,6,7,8,9\right\}$

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