Consider rolling a six-sided die. Let $A$ be the set of outcomes where the roll is an even number. Let $B$ be the set of outcomes where the roll is greater than 3 . Calculate and compare the sets on both sides of De Morgan’s laws
$$
(A \cup B)^{r}=A^{c} \cap B^{c} . \quad(A \cap B)^{c}=A^{c} \cup B^{c}
$$
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We have
$$
A=\{2,4,6\}, \quad B=\{4,5,6\},
$$
so $A \cup B=\{2,4,5,6\}$, and
$$
(A \cup B)^{c}=\{1,3\} .
$$
On the other hand,
$$
A^{c} \cap B^{c}=\{1,3,5\} \cap\{1,2,3\}=\{1,3\}
$$
Similarly, we have $A \cap B=\{4,6\}$, and
$$
(A \cap B)^{c}=\{1,2,3,5\}
$$
On the other hand,
$$
A^{c} \cup B^{c}=\{1,3,5\} \cup\{1,2,3\}=\{1,2,3,5\}
$$
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