# Key for Learning Target 8 Quiz 3

## :new: Learning Target 8

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(**CORE**) I can find the cardinality, power set, and complement of a set; and I can find the intersection, union, difference, and Cartesian product of two sets.
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Consider the sets:
$$A = \{1,2\} \qquad B = \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 3}\} \qquad C = \{2,4,6,8\}$$
The universal set $U$ is the set of all natural numbers.
Determine each of the following. If the answer is a set, write it in roster notation.

1. $|C \setminus A| = 3$

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This is asking for the *cardinality* of the set $C \setminus A$. That set is $C \setminus A = \{4,6,8\}$.
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2. $A \cup C = \{1,2,4,6,8\}$
3. $A \cap B = \emptyset$
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This is asking for the intersection of $A$ and $B$, i.e. the set of all elements that belong to both sets. But neither 1 nor 2 is a multiple of 3, so that set is empty.
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4. $A \times \{u,v\} = \{ (1,u), (1,v), (2,u), (2,v)\}$
5. ${\cal{P}} (A)$ (the power set of $A$)

$${\cal{P}}(A) = \{ \emptyset, \{1\}, \{2\}, \{1,2\}\}$$

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The power set of $A$ is the set whose elements are all of the subsets of $A$. Please note $\emptyset$ is a subset of $A$, but $\{ \emptyset\}$ is not.
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6. $\overline{B}$ (the complement of $B$)

$$\overline{B} = \mathbb{N} \setminus \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 3}\} \\
= \{1,2,4,5,7,8,10,11,13,14,\dots\}$$

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The complement of this set is the set of all natural numbers that are *not* multiples of 3. **Note that the Success Criteria stated on the quiz says that all sets must be given in roster notation** so you cannot just give $\{n \in \mathbb{N} \, : \, n \ \text{is NOT a multiple of 3}\}$ as a response.

And note, 0 is a multiple of 3 since $0 = 0 \times 3$.
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