Number Sets & Intervals — Practice Quiz

A Set Theory cheat sheet for Number Sets & Intervals — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Natural numbers: the counting numbers (some texts include 0)

$$\mathbb{N} = \{1, 2, 3, \dots\}$$

Integers: the whole numbers and their negatives

$$\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$$

Rational numbers: ratios of integers

$$\mathbb{Q} = \left\{\, \tfrac{p}{q} \mid p, q \in \mathbb{Z},\ q \neq 0 \,\right\}$$

Real numbers: every point on the number line (rationals and irrationals together)

$$\mathbb{R}$$

The number systems nest: each is a subset of the next

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$

Closed interval: includes both endpoints

$$[a, b] = \{\, x \in \mathbb{R} \mid a \le x \le b \,\}$$

Open interval: excludes both endpoints

$$(a, b) = \{\, x \in \mathbb{R} \mid a < x < b \,\}$$

Half-open interval: includes a, excludes b

$$[a, b) = \{\, x \in \mathbb{R} \mid a \le x < b \,\}$$

Unbounded interval: all reals greater than a

$$(a, \infty) = \{\, x \in \mathbb{R} \mid x > a \,\}$$

Practice quiz

  1. Which of the following numbers is an integer but not a natural number?

    • $5$
    • $0$
    • $-3$
    • $\frac{1}{2}$

    Answer: $-3$

  2. Which of the following statements correctly describes the relationship between number sets?

    • $\mathbb{Z} \subset \mathbb{N}$
    • $\mathbb{Q} \subset \mathbb{Z}$
    • $\mathbb{R} \subset \mathbb{Q}$
    • $\mathbb{N} \subset \mathbb{Q}$

    Answer: $\mathbb{N} \subset \mathbb{Q}$

  3. Which of the following numbers is a rational number?

    • $\sqrt{2}$
    • $\pi$
    • $0.333...$
    • $e$

    Answer: $0.333...$

  4. All of the following numbers are real numbers EXCEPT:

    • $7$
    • $-1.5$
    • $\sqrt{9}$
    • $\sqrt{-4}$

    Answer: $\sqrt{-4}$

  5. Which of the following numbers is included in the closed interval $[2, 5]$?

    • $1.9$
    • $5.1$
    • $2$
    • $5.000001$

    Answer: $2$

  6. The open interval $( -3, 7 )$ represents all real numbers $x$ such that:

    • $x \ge -3$ and $x \le 7$
    • $x > -3$ and $x < 7$
    • $x \ge -3$ and $x < 7$
    • $x > -3$ and $x \le 7$

    Answer: $x > -3$ and $x < 7$

  7. Which interval notation correctly represents the set of all real numbers $x$ such that $0 < x \le 10$?

    • $[0, 10]$
    • $(0, 10)$
    • $[0, 10)$
    • $(0, 10]$

    Answer: $(0, 10]$

  8. The set of all real numbers $x$ such that $x > -5$ can be written in interval notation as:

    • $[-5, \infty)$
    • $(-5, \infty)$
    • $(- \infty, -5)$
    • $(- \infty, -5]$

    Answer: $(-5, \infty)$

  9. Consider the number $x = \frac{1}{4}$. To which of the following sets does $x$ belong?

    • $\mathbb{N}$ only
    • $\mathbb{Z}$ only
    • $\mathbb{Q}$ only
    • $\mathbb{R}$ and $\mathbb{Q}$

    Answer: $\mathbb{R}$ and $\mathbb{Q}$

  10. A student's score $S$ on a test must be greater than $70$ and less than or equal to $90$. Which interval represents the possible scores?

    • $[70, 90]$
    • $(70, 90)$
    • $[70, 90)$
    • $(70, 90]$

    Answer: $(70, 90]$

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