Define sets A and B and find their intersection and union. Set A contains all positive integers. Set B contains all negative integers.

Set A is defined as the set of all $x$ such that $x$ is a positive integer.

$$A = \{x / x \in \mathbb{Z}^+\}$$

This means A includes all integers greater than 0.

$$A = \{1, 2, 3, 4, \dots\}$$

Set B is defined as the set of all $x$ such that $x$ is an integer and $x$ is less than 0.

$$B = \{x / x \in \mathbb{Z} \text{ and } x < 0\}$$

This means B includes all negative integers.

$$B = \{\dots, -4, -3, -2, -1\}$$

The intersection of two sets contains elements that are common to both sets.
Set A contains only positive integers. Set B contains only negative integers. There are no common elements between positive and negative integers.

$$A \cap B = \emptyset$$

The intersection of A and B is the empty set.

$$\boxed{A \cap B = \emptyset}$$

The union of two sets contains all elements that are in either set (or both).
Combining the elements of A and B:

$$A \cup B = \{\dots, -3, -2, -1, 1, 2, 3, \dots\}$$

This set includes all integers except for 0. This can be written as the set of all integers excluding 0.

$$\boxed{A \cup B = \mathbb{Z} \setminus \{0\}}$$