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If the universal set is given by $S=\{1,2,3,4,5,6\}$, and $A=\{1,2\}, B=\{2,4,5\}, C=\{1,5,6\}$ are three sets, find the following sets:

a. $A \cup B$
b. $A \cap B$
c. $\bar{A}$
d. $\bar{B}$
e. Check De Morgan's law by finding $(A \cup B)^{c}$ and $A^{c} \cap B^{c}$.
f. Check the distributive law by finding $A \cap(B \cup C)$ and $(A \cap B) \cup(A \cap C)$.
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a. $A \cup B=\{1,2,4,5\}$.
b. $A \cap B=\{2\}$.
c. $\bar{A}=\{3,4,5,6\}$ ( $\bar{A}$ consists of elements that are in $S$ but not in $A$ ).
d. $\bar{B}=\{1,3,6\}$.
e. We have
$$(A \cup B)^{c}=\{1,2,4,5\}^{c}=\{3,6\}$$
which is the same as
$$A^{c} \cap B^{c}=\{3,4,5,6\} \cap\{1,3,6\}=\{3,6\}$$
f. We have
$$A \cap(B \cup C)=\{1,2\} \cap\{1,2,4,5,6\}=\{1,2\}$$
which is the same as
$$(A \cap B) \cup(A \cap C)=\{2\} \cup\{1\}=\{1,2\} .$$
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