Question

Find the following cardinalities. $$ \left|\left\\{x \in \mathbb{Z}: x^{2}<10\right\\}\right| $$

Short answer

The cardinality of the set is 7.

Step-by-step solution

Identify the Possible Values

First, identify the integers that satisfy the condition \(x^{2}<10\). This means looking for values where the square is less than 10. Examine both positive and negative integers, because the square of a negative integer is also positive.

Enumerate the Values

Write down the integers that satisfy this condition. Remember that zero is neither positive or negative, so it should also be considered. The integers that satisfy \(x^{2}<10\) are -3, -2, -1, 0, 1, 2, 3.

Determine the Cardinality

Lastly, count the number of elements that were identified in step 2. There are seven integers that satisfy this condition (-3, -2, -1, 0, 1, 2, 3). So, the cardinality of the set is 7.