Discrete & Continuous Sets — Hard Practice Quiz

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

Practice quiz

1. A set $S$ satisfies the condition $\forall\, a < b,\ \exists\, c \in S:\ a < c < b$ and also $|S| = |\mathbb{N}|$. Which of the following statements is true regarding $S$?

   • S must be a continuous set.
   • S must be an uncountable set.
   • S is an example of a discrete set.
   • S is isomorphic to the set of rational numbers $\mathbb{Q}$.

   Answer: S is isomorphic to the set of rational numbers $\mathbb{Q}$.

2. Consider a set $A$ such that $|A| = \aleph_0$. If $A$ is also a continuous set, what can be concluded about $A$?

   • A must be a finite set.
   • A must be an uncountable set.
   • Such a set $A$ cannot exist.
   • A must be a discrete set.

   Answer: Such a set $A$ cannot exist.

3. If a set $S$ is uncountable, which of the following statements must be false?

   • $S$ is an infinite set.
   • $S$ is a dense set.
   • $S$ is a discrete set.
   • $S$ is a continuous set.

   Answer: $S$ is a discrete set.

4. A set $X$ is defined by the property that between any two distinct elements $a, b \in X$, there exists an element $c \in X$ where $a < c < b$. Additionally, $X$ cannot be put into a one-to-one correspondence with the natural numbers $\mathbb{N}$. Which of the following best describes $X$?

   • $X$ is a countably infinite and discrete set.
   • $X$ is an uncountable and continuous set.
   • $X$ is a finite and dense set.
   • $X$ is a discrete set with cardinality $\aleph_0$.

   Answer: $X$ is an uncountable and continuous set.

5. Let $S$ be a set where $|S| = \infty$. If $S$ is also a discrete set, what can be concluded about its cardinality?

   • $|S| < \aleph_0$
   • $|S| = \aleph_0$
   • $|S| = 2^{\aleph_0}$
   • The cardinality cannot be determined from the given information.

   Answer: $|S| = \aleph_0$

6. Consider the set of all irrational numbers, denoted as $\mathbb{R} \setminus \mathbb{Q}$. Which combination of properties from the given formulas accurately describes this set?

   • Finite and discrete.
   • Countably infinite and continuous.
   • Uncountable and dense.
   • Discrete and uncountable.

   Answer: Uncountable and dense.

7. A set $A$ is defined by the property that for any $a, b \in A$ with $a < b$, there exists $c \in A$ such that $a < c < b$. If $A$ is also known to be countably infinite, which of the following sets could $A$ represent?

   • The set of natural numbers $\mathbb{N}$.
   • The set of integers $\mathbb{Z}$.
   • The set of rational numbers $\mathbb{Q}$.
   • The set of real numbers $\mathbb{R}$.

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

8. If a set $S$ is continuous, which of the following statements about $S$ must be true?

   • $S$ is a discrete set.
   • $S$ is a countably infinite set.
   • $S$ is an uncountable and dense set.
   • $S$ is a finite set.

   Answer: $S$ is an uncountable and dense set.

9. A set $P$ is such that its elements can be listed in a sequence $p_1, p_2, p_3, \dots$ without repetition, and for any two distinct elements $x, y \in P$, there is a positive minimum distance $\delta > 0$ such that $|x-y| \ge \delta$. Which of the following best describes $P$?

   • $P$ is a dense and uncountable set.
   • $P$ is a continuous set.
   • $P$ is a countably infinite and discrete set.
   • $P$ is a finite set.

   Answer: $P$ is a countably infinite and discrete set.

10. Consider a set $M$ that is a subset of $\mathbb{R}$. If $M$ is dense in $\mathbb{R}$ but not continuous, what can be concluded about its cardinality?

    • $M$ must be countably infinite.
    • $M$ must be uncountable.
    • $M$ must be a finite set.
    • $M$ must be an infinite set, but its countability cannot be determined.

    Answer: $M$ must be an infinite set, but its countability cannot be determined.