Decide whether the statement Every integer is a rational number is true or false under the standard mathematical definitions of integers and rational numbers.

Introduction / Context:
This question checks basic number theory and the nested structure of number sets. It asks whether every integer belongs to the set of rational numbers. Understanding the formal definition of a rational number and how integers fit into that definition is essential in algebra and higher mathematics, because many properties and theorems depend on these set relationships.

Given Data / Assumptions:
- An integer is any whole number, positive, negative, or zero, such as ..., -3, -2, -1, 0, 1, 2, 3, ....
- A rational number is any number that can be written in the form p / q where p and q are integers and q is not zero.
- Standard school-level definitions are used without any non-standard conventions.

Concept / Approach:
To test whether every integer is a rational number, we need to see whether an arbitrary integer can always be represented in the form p / q with integer p and q and q not equal to zero. If this is possible for all integers, then the statement is true. If even one integer cannot be written in that fractional form, the statement would be false. We therefore try to represent a general integer n in rational form.

Step-by-Step Solution:
Step 1: Let n be any integer. It may be positive, negative, or zero. Step 2: Consider the fraction n / 1. Here the numerator n is an integer, and the denominator 1 is also an integer. Step 3: The denominator 1 is not zero, so n / 1 satisfies the requirement for being a rational number. Step 4: The fraction n / 1 is equal to n itself, so every integer n can be written as a rational number without changing its value. Step 5: Since n was arbitrary, this reasoning holds for all integers, including negative integers and zero.

Verification / Alternative check:
Try specific examples: 5 can be written as 5 / 1; -3 can be written as -3 / 1; 0 can be written as 0 / 1. In each case, numerator and denominator are integers and the denominator is not zero. Therefore, 5, -3, and 0 are all rational numbers. This confirms that the set of integers is fully contained within the set of rational numbers.

Why Other Options Are Wrong:
Option FALSE: This would require at least one integer to fail the rational number definition, which does not happen.
Option Cannot be determined: The definitions are precise, so there is no ambiguity; the statement’s truth can be determined exactly.
Option True only for positive integers: Negative integers and zero also fit the form n / 1, so limiting to positive integers is unnecessary and incorrect.
Option True only for non-zero integers: Zero is 0 / 1 and is rational, so excluding zero is not justified.

Common Pitfalls:
Some learners incorrectly think rational means fraction that is not a whole number, and therefore wrongly exclude integers. Others confuse rational with real or with decimal numbers that terminate or repeat. Remember that rational specifically means expressible as a ratio of two integers with a non-zero denominator. Under this definition, every integer is automatically rational.

Final Answer:
The statement Every integer is a rational number is TRUE.