Question

Find the following cardinalities. $$ |\\{x \in \mathbb{Z}:|x|<10\\}| $$

Short answer

The cardinality of the set is 19.

Step-by-step solution

Define the Range

First, interpret the definition of the set. The absolute value \(|x|\) is less than 10. This means that \(x\) can be any integer from -9 to 9.

Enumerate the Elements

Once the range is defined, list out the integers that fall within that range. These are: \(-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\).

Count the Elements

Now that all the elements of the set have been listed, count them to find the cardinality of the set. Counting the numbers listed in step 2, there are 19 integers.