Subsets of real numbers worksheet - Free Printable

Let's go through each statement one by one and determine whether it is TRUE or FALSE, with explanations.

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1. All counting numbers are integers.

✔ TRUE
- Counting numbers (also called natural numbers) are: 1, 2, 3, 4, ...
- These are all positive whole numbers and are included in the set of integers (..., -2, -1, 0, 1, 2, ...).
✔️ So yes, all counting numbers are integers.

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2. The integer 8 can be expressed as a fraction.

✔ TRUE
- Any integer can be written as a fraction with denominator 1.
- For example: $ 8 = \frac{8}{1} $.
✔️ So yes, 8 can be expressed as a fraction.

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3. Non-repeating, non-terminating decimals are irrational.

✔ TRUE
- Irrational numbers cannot be written as fractions and have decimal expansions that are non-repeating and non-terminating.
- Examples: $ \pi = 3.14159... $, $ \sqrt{2} = 1.41421... $
✔️ So this definition is correct.

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4. Whole numbers exclude zero.

✘ FALSE
- Whole numbers include zero and all positive integers: 0, 1, 2, 3, ...
- Counting numbers start from 1, but whole numbers include 0.
✖️ So this statement is false.

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5. The number 3.35 is a rational number.

✔ TRUE
- A rational number is any number that can be written as a fraction $ \frac{p}{q} $ where $ p $ and $ q $ are integers and $ q \ne 0 $.
- $ 3.35 = \frac{335}{100} = \frac{67}{20} $ → a fraction of two integers.
✔️ So it is rational.

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6. Irrational numbers are not included in the number line.

✘ FALSE
- All real numbers, including irrational numbers, are located on the number line.
- For example, $ \sqrt{2} $, $ \pi $, etc., have positions on the number line.
✖️ So this is false — irrational numbers are on the number line.

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7. The largest negative integer is −1

✔ TRUE
- Negative integers: ..., -3, -2, -1
- Among these, -1 is the closest to zero, so it is the largest (least negative).
✔️ So yes, -1 is the largest negative integer.

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8. Zero is a positive integer

✘ FALSE
- Zero is neither positive nor negative.
- Positive integers are: 1, 2, 3, ...
- Zero is an integer, but not positive.
✖️ So this is false.

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9. The union of the set of rational numbers and the set of irrational numbers is the set of real numbers.

✔ TRUE
- Every real number is either rational or irrational — no overlap.
- Their union covers all real numbers.
✔️ So yes, this is correct.

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10. The intersection of the set of positive integers and the set of negative integers is an empty set.

✔ TRUE
- Positive integers: 1, 2, 3, ...
- Negative integers: -1, -2, -3, ...
- There is no number that is both positive and negative.
- So their intersection is empty.
✔️ Therefore, this is true.

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✔ Final Answers:

| Statement | Answer |
|--------|--------|
| 1 | TRUE |
| 2 | TRUE |
| 3 | TRUE |
| 4 | FALSE |
| 5 | TRUE |
| 6 | FALSE |
| 7 | TRUE |
| 8 | FALSE |
| 9 | TRUE |
| 10 | TRUE |

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