Lesson 1 — Number sets, intervals, notation

List, in brace notation, the set $A = \{x \in \mathbb{N} : 1 \leq x \leq 5\}$.

Write in interval notation: $\{x \in \mathbb{R} : 2 \leq x \leq 8\}$.

Write in interval notation: $\{x \in \mathbb{R} : x < 5\}$.

Given $A = \{1, 2, 3, 4, 5\}$ and $B = \{4, 5, 6, 7\}$, compute $A \cap B$.

Using the same $A$ and $B$ from the previous exercise, compute $A \cup B$.

Still with the same $A, B$: compute $A \setminus B$ (elements in $A$ but not in $B$).

Compute $[3, 10] \cap (1, 7)$.

Compute $[3, 10] \cup (1, 7)$.

$(-\infty, 0] \cup [0, +\infty) = ?$

$(-\infty, 0] \cap [0, +\infty) = ?$

True or false: $\mathbb{N} \subseteq \mathbb{Z}$. (Use T or F.)

True or false: $\mathbb{Z} \subseteq \mathbb{N}$.

True or false: $\sqrt{2} \in \mathbb{R}$ but $\sqrt{2} \notin \mathbb{Q}$.

Solve and express as an interval: $|2x - 3| \leq 5$.

Solve and express as an interval: $|x| > 2$.

Solve: $2 \leq |x| \leq 3$.

Show that if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Let $A = (1, 5)$ and $B = [2, 7)$. Find $A \cap B$ and $A \cup B$. Also represent both on a number line.

Simplify: $(A^c \cap B^c)^c$.

Let $A = \{x \in \mathbb{R} : x^2 \leq 4\}$ and $B = \{x \in \mathbb{R} : x \geq 0\}$. Determine $A \cap B$ in interval notation.

A regulatory agency classifies an engine as inefficient if its efficiency $\eta$ is less than $30\%$, average if $30\% \leq \eta < 70\%$, and efficient if $\eta \geq 70\%$. Express each range in interval notation (with $\eta \in [0, 1]$).

A medication leaflet states a pediatric dose of $0.5 \leq m \leq 12.5$ mg/kg of body weight. For a child weighing 30 kg, what is the recommended total dose interval in mg?

You are programming a thermostat. It turns the heater on when temperature $T < 18\,°C$ and turns it off when $T \geq 22\,°C$. (a) Determine the interval of "temperatures at which the heater is on" as a function of measured $T$. (b) Note that the interval $[18, 22)$ is "ambiguous" — explain what happens with the thermostat hysteresis.

In a survey, $80\%$ of people read newspaper A, $40\%$ read B, and $30\%$ read both. What percentage reads at least one of the newspapers? And what percentage reads neither?

In a class of 100 students: 50 take math, 30 physics, 25 chemistry. 10 take both math and physics, 8 both math and chemistry, 5 both physics and chemistry, and 3 take all three. How many students take none of the three?

Solve: $|x - 1| + |x + 2| \leq 5$. Express the answer as an interval.

Classic proof. Prove that $\sqrt{2}$ is irrational.

Prove one of De Morgan's laws: $(A \cup B)^c = A^c \cap B^c$.

Show that between any two distinct rationals there is another rational. (Density of ℚ.)

How many integers belong to the set $\{x \in \mathbb{R} : x^2 - 5x + 6 \leq 0\}$?

Solve $|x - 1| \leq 5$.

Solve $|x - 1| \geq 2$.

Solve $|2x - 4| < 6$.

Solve $|3x + 1| > 7$.

Solve $|x + 2| < -3$.

Solve $|x + 5| > -1$.

Let $A = (-\infty, 4]$ and $B = [-2, 6)$. Determine $A \cap B$ and $A \cup B$ in interval notation.

Let $A = [1, 8]$, $B = (3, 12]$, and $C = (-\infty, 5]$. Compute $(A \cap B) \cup C$.

Express in set notation: "all reals greater than $-3$ and less than or equal to $7$, except $2$".

Solve $|2x - 1| + 3 \leq 8$ and express as an interval.

Solve the system $\begin{cases} x \geq 0 \\ |x - 4| \leq 2 \end{cases}$.

Solve the system $\begin{cases} -3 \leq x \leq 5 \\ x^2 \geq 4 \end{cases}$. Express the solution as a union of intervals.

True or false: $|a - b| = |b - a|$ for all $a, b \in \mathbb{R}$. Justify without using numerical values.

Show that $|a + b| \leq |a| + |b|$ (triangle inequality) by testing with (a) $a = 3, b = -5$; (b) $a = -2, b = 4$; (c) $a = 0, b = -1$.

Determine the set $\{x \in \mathbb{Z} : -3 < x \leq 4\}$ by enumeration.

An industrial facility checks the output voltage of a piece of equipment with a tolerance of $\pm 0.3$ V around $12$ V. Express the acceptable range $V$ as an interval.

Healthy blood pressure in adults is classified (SBC, 2025) as optimal when systolic pressure is less than $120$ mmHg and diastolic pressure is less than $80$ mmHg. Express "optimal pressure" as a subset of the Cartesian product $\mathbb{R} \times \mathbb{R}$.

On an industrial scale, parts with mass in $[995, 1\,005]$ g are considered within spec. Express "out of spec" as a union of intervals.

In statistical process control, UCL (upper control limit) and LCL (lower control limit) are defined for a variable $X$. The process is in control if $X \in [\text{LCL}, \text{UCL}]$. For $\text{LCL} = 9.7$ kg and $\text{UCL} = 10.3$ kg, determine whether the measurements $X = 9.65$, $X = 10.1$, $X = 10.35$ are within control.

A cooperative pays R$ 1.80/L of milk up to 1,000 L/month; between 1,000 and 5,000 L it pays R$ 2.00/L; above 5,000 L it pays R$ 2.30/L. Model the payment $P(q)$ as a piecewise function defined on $\{q \in \mathbb{R} : q \geq 0\}$. (This function will return as a piecewise linear function in Lesson 3.)

The acceptable pH range for drinking water is $[6{,}0, 9{,}5]$. Consider $9$ samples with pH: $5{,}8 \mid 6{,}1 \mid 6{,}5 \mid 7{,}0 \mid 7{,}3 \mid 8{,}9 \mid 9{,}5 \mid 9{,}8 \mid 10{,}2$. How many samples are within the range? Express as a set.

On a production scale, the measurement error $\epsilon$ satisfies $|\epsilon| \leq 0{,}5\%$ of the reading. For a reading of $200{,}0$ g, what interval contains the true value?

A consumer GPS has accuracy $\pm 3$ m under normal conditions. If the device shows coordinate $x = 47{,}25$ m, describe the uncertainty range as an interval.

The recommended maximum heart rate for aerobic exercise is $0{,}85 \cdot (220 - \text{age})$. For a 30-year-old, express the "light to moderate exercise" range as $[60\%,\ 75\%]$ of that heart rate. Compute numerically.

For an electronic circuit to operate correctly, the supply voltage $V$ must satisfy: $V \geq 4{,}5$ V and $V \leq 5{,}5$ V and $V \neq 5{,}0$ V (regulator limitation). Express the set of acceptable voltages as a union of disjoint intervals.

Show that among any 11 integers between 1 and 20, there are always two that differ by exactly 5. (Pigeonhole Principle.)

The Cantor set is constructed by recursively removing the middle third of $[0, 1]$. After $n$ steps, what is the total length of the remaining intervals? What value does this length approach as $n \to \infty$?

Prove that if $A \subseteq B$, then $A \cup B = B$ and $A \cap B = A$.

Prove that $\sqrt{3}$ is irrational. (Adapt the proof for $\sqrt{2}$.)

Prove the other De Morgan law: $(A \cap B)^c = A^c \cup B^c$.