Discrete & Continuous Sets — Practice Quiz

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

Formulas & key concepts

Finite set: a limited number of elements. Example: {red, green, blue}

$$|A| < \infty$$

Infinite set: the elements never run out. Example: the naturals ℕ

$$|A| = \infty$$

Discrete set: separated, isolated points you can list one by one, with gaps between them (like ℤ)

$$\{\dots, -2, -1, 0, 1, 2, \dots\}$$

Continuous set: an unbroken range with no gaps between values (like the interval [0, 1])

$$[a, b] \subset \mathbb{R}$$

Countably infinite: can be paired one-to-one with the naturals. ℤ and ℚ are countable

$$|\mathbb{N}| = \aleph_0$$

Uncountable: too large to list even in principle — no one-to-one pairing with ℕ. The reals ℝ are uncountable

$$|\mathbb{R}| = 2^{\aleph_0}$$

Dense set: between any two elements lies another. ℚ and ℝ are dense; ℤ is not

$$\forall\, a < b,\ \exists\, c \in S:\ a < c < b$$

Practice quiz

  1. Which of the following sets is finite?

    • The set of all prime numbers.
    • The set of all even integers.
    • The set of all solutions to $x^2 = 4$.
    • The set of all points on a line segment.

    Answer: The set of all solutions to $x^2 = 4$.

  2. Which of the following sets is discrete?

    • The interval $[0, 1]$
    • The set of rational numbers $\mathbb{Q}$
    • The set of natural numbers $\mathbb{N}$
    • The set of real numbers $\mathbb{R}$

    Answer: The set of natural numbers $\mathbb{N}$

  3. According to the formula $|\mathbb{N}| = \aleph_0$, which of the following sets is countably infinite?

    • The set of all real numbers between $0$ and $1$.
    • The set of all points in a square.
    • The set of all rational numbers $\mathbb{Q}$.
    • The set of all irrational numbers.

    Answer: The set of all rational numbers $\mathbb{Q}$.

  4. The set of real numbers $\mathbb{R}$ is described by the formula $|\mathbb{R}| = 2^{\aleph_0}$. What does this imply about $\mathbb{R}$?

    • It is a finite set.
    • It is countably infinite.
    • It is uncountable.
    • It is a discrete set.

    Answer: It is uncountable.

  5. Which of the following sets is dense, meaning that for any two distinct elements $a$ and $b$ in the set, there exists an element $c$ such that $a < c < b$?

    • The set of natural numbers $\mathbb{N}$.
    • The set of integers $\mathbb{Z}$.
    • The set of rational numbers $\mathbb{Q}$.
    • The set $\{1, 2, 3, 4, 5\}$.

    Answer: The set of rational numbers $\mathbb{Q}$.

  6. An interval like $[a, b] \subset \mathbb{R}$ is an example of what type of set?

    • Discrete
    • Countably infinite
    • Continuous
    • Finite

    Answer: Continuous

  7. Consider the set of all possible outcomes when rolling a standard six-sided die once. Which statement is true about this set?

    • It is an infinite set.
    • Its cardinality is $\aleph_0$.
    • It is a finite set.
    • It is an uncountable set.

    Answer: It is a finite set.

  8. Which of the following statements correctly distinguishes between countably infinite and uncountable sets?

    • Countably infinite sets can be put into one-to-one correspondence with $\mathbb{R}$, while uncountable sets cannot.
    • Uncountable sets are always finite, while countably infinite sets are not.
    • Countably infinite sets can be put into one-to-one correspondence with $\mathbb{N}$, while uncountable sets cannot.
    • Uncountable sets are discrete, while countably infinite sets are continuous.

    Answer: Countably infinite sets can be put into one-to-one correspondence with $\mathbb{N}$, while uncountable sets cannot.

  9. Based on the provided formulas, which of the following properties applies to the set of integers $\mathbb{Z}$?

    • It is continuous.
    • It is uncountable.
    • It is dense.
    • It is discrete.

    Answer: It is discrete.

  10. The set of rational numbers $\mathbb{Q}$ has which of the following properties?

    • It is uncountable and discrete.
    • It is countably infinite and dense.
    • It is finite and continuous.
    • It is uncountable and continuous.

    Answer: It is countably infinite and dense.

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