Number Sets & Intervals — Hard Practice Quiz

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

Practice quiz

1. Let $A = [-5, 3)$ and $B = (0, 7]$. Consider the set $C = A \cap B$. Which of the following statements is true regarding the elements of $C$?

   • The set $C$ contains exactly $2$ natural numbers.
   • The smallest integer in $C$ is $0$.
   • All elements of $C$ are rational numbers.
   • $C$ can be expressed as $\{\, x \in \mathbb{Z} \mid 0 < x < 3 \,\}$.

   Answer: The set $C$ contains exactly $2$ natural numbers.

2. Let $P = (1, \infty)$ and $Q = [2, 5]$. Which of the following statements is false?

   • $Q \subset P$.
   • All elements of $Q$ are rational numbers.
   • The set $Q$ contains exactly $4$ integers.
   • The set $P$ contains infinitely many natural numbers.

   Answer: All elements of $Q$ are rational numbers.

3. Consider the interval $I = (-1, 2]$. Which of the following statements is true?

   • The set of integers in $I$ is $\{-1, 0, 1, 2\}$.
   • The set of natural numbers in $I$ is $\{1, 2\}$.
   • Every rational number $q$ such that $q \in I$ satisfies $q \in \mathbb{Z}$.
   • The interval $I$ contains no irrational numbers.

   Answer: The set of natural numbers in $I$ is $\{1, 2\}$.

4. Let $S = [0, 1)$. Which of the following statements is true?

   • $S$ contains exactly one natural number.
   • $S$ contains infinitely many integers.
   • Every element $x \in S$ is also an element of $\mathbb{Q}$.
   • If $y \in S$ and $y \in \mathbb{Q}$, then $y$ can be written as $\frac{p}{q}$ where $p \in \mathbb{Z}$ and $q \in \mathbb{N}$.

   Answer: If $y \in S$ and $y \in \mathbb{Q}$, then $y$ can be written as $\frac{p}{q}$ where $p \in \mathbb{Z}$ and $q \in \mathbb{N}$.

5. Let $K = (0, \infty)$. Which of the following statements is false?

   • $K$ contains infinitely many integers.
   • Every natural number is an element of $K$.
   • The intersection of $K$ with the set of non-positive real numbers is the empty set.
   • Every element $x \in K$ such that $x \in \mathbb{Q}$ implies $x \in \mathbb{N}$.

   Answer: Every element $x \in K$ such that $x \in \mathbb{Q}$ implies $x \in \mathbb{N}$.

6. Let $A = [-2, 5]$ and $B = (1, 4)$. Consider the set $D = A \setminus B$ (elements in $A$ but not in $B$). How many integers are in $D$?

   • $4$
   • $5$
   • $6$
   • $7$

   Answer: $6$

7. Let $X = [0, 2)$ and $Y = (1, \infty)$. Which of the following statements is true about the set $Z = X \cap Y$?

   • $Z$ contains exactly one integer.
   • $Z$ contains infinitely many rational numbers.
   • $Z$ is a subset of $\mathbb{N}$.
   • The smallest real number in $Z$ is $1$.

   Answer: $Z$ contains infinitely many rational numbers.

8. Consider the interval $J = (0, 1)$. Which of the following statements is true?

   • $J$ contains no rational numbers.
   • $J$ contains no integers.
   • $J$ contains all natural numbers.
   • $J$ is a subset of $\mathbb{Z}$.

   Answer: $J$ contains no integers.

9. Let $M = [0, 1]$. Consider the set $R_M = \{\, x \in \mathbb{Q} \mid x \in M \,\}$. Which of the following statements is true about $R_M$?

   • $R_M$ contains only integers.
   • Every element $x \in R_M$ can be written as $\frac{p}{q}$ where $p \in \mathbb{Z}$, $q \in \mathbb{N}$, and $p \le q$.
   • $R_M$ contains no irrational numbers.
   • $R_M$ is a finite set.

   Answer: Every element $x \in R_M$ can be written as $\frac{p}{q}$ where $p \in \mathbb{Z}$, $q \in \mathbb{N}$, and $p \le q$.

10. Let $A = [-1, 3]$, $B = (2, 5)$, and $C = (0, \infty)$. Which of the following statements is true?

    • $A \cap B = (2, 3]$.
    • $B \subset C$.
    • $A \cup B = [-1, 5)$.
    • All of the above.

    Answer: All of the above.