Using the symbols , , , and , determine the relationship between the given sets and elements for Exercise 1, Question 1.

5 is an integer, and $\mathbb{Z}$ is the set of all integers. Therefore, 5 is an element of $\mathbb{Z}$.

$$5 \in \mathbb{Z}$$

$P(\mathbb{Z})$ is the power set of $\mathbb{Z}$, meaning its elements are subsets of $\mathbb{Z}$. 5 is an integer, not a set, so it cannot be an element of $P(\mathbb{Z})$.

$$5 \notin P(\mathbb{Z})$$

$\{5\}$ is a set containing the integer 5. Since 5 is an element of $\mathbb{Z}$, the set $\{5\}$ is a subset of $\mathbb{Z}$.

$$\{5\} \subset \mathbb{Z}$$

As established in Step 3, $\{5\}$ is a subset of $\mathbb{Z}$. The elements of $P(\mathbb{Z})$ are precisely the subsets of $\mathbb{Z}$. Therefore, $\{5\}$ is an element of $P(\mathbb{Z})$.

$$\{5\} \in P(\mathbb{Z})$$

For $\{-2, \frac{1}{4}\}$ to be an element of $P(\mathbb{Z})$, it must be a subset of $\mathbb{Z}$. The element $\frac{1}{4}$ is not an integer ($\frac{1}{4} \notin \mathbb{Z}$). Since not all elements of $\{-2, \frac{1}{4}\}$ are in $\mathbb{Z}$, the set $\{-2, \frac{1}{4}\}$ is not a subset of $\mathbb{Z}$. Consequently, it cannot be an element of $P(\mathbb{Z})$.

$$\{-2, \frac{1}{4}\} \notin P(\mathbb{Z})$$

Here are the completed statements:

1. $5 \boxed{\in} \mathbb{Z}$
2. $5 \boxed{\notin} P(\mathbb{Z})$
3. $\{5\} \boxed{\subset} \mathbb{Z}$
4. $\{5\} \boxed{\in} P(\mathbb{Z})$
5. $\{-2, \frac{1}{4}\} \boxed{\notin} P(\mathbb{Z})$