Mathematics · Module 1

16 lessons · 300 problems

Sets & Inequalities — full chapter

Learn each idea, drill it with Basic → Intermediate → Advanced problems, and finish with mixed practice. Every problem shows its thinking right after you answer.

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Lessons Formulas Traps Practice 100 Previous exams Checklist

After this chapter you can

Read and write sets in roster, set-builder, and descriptive form.
Count subsets ( $2^n$ ) and proper subsets ( $2^n-1$ ).
Compute union, intersection, difference, complement, and Cartesian products.
Apply De Morgan's laws and inclusion–exclusion (2-set and 3-set).
Solve linear, compound, absolute-value, quadratic, polynomial, and rational inequalities.
Run sign charts and always exclude denominator zeros.
Translate “at least / at most / only / neither” into exact math.

How to study this chapter

Read the concept in simple words.
Write down the exact notation.
Work one easy example by hand.
Memorize the general formula or method.
Solve one exam-style example.
Name the trap for this type out loud.
Solve 2–5 questions: Basic → Intermediate → Advanced.
Finish with mixed practice — identify the method yourself first.
Ask: what type is this? → what rule? → fastest safe method? → boundaries checked?

Lessons

Lesson 1 of 16

Set Basics

Learn the idea

A **set** is a well-defined collection of distinct objects, called **elements**. Write

x \in A

when

x

belongs to

A

, and

x \notin A

when it does not.

Never confuse the two:

x \in A

says

x

is an *element* of

A

, while

\{x\} \subseteq A

says the *set containing*

x

is a subset of

A

Key rules to memorize

Finite sets list elements: $A = \{2, 4, 6, 8\}$ , so $|A| = 4$ .
Special sets: empty $\varnothing$ , singleton $\{5\}$ , infinite $\mathbb{N} = \{1, 2, 3, \ldots\}$ .
Sets are equal iff they hold exactly the same elements — order and repetition don't matter: $\{1, 2, 3\} = \{3, 2, 1\}$ .

Worked example

Let $A = \{1, 2, 3, 4\}$ . Which are true: $3 \in A$ , $5 \in A$ , $\{3\} \subseteq A$ ?

$3$ is listed in $A$ , so $3 \in A$ is true.
$5$ is not listed, so $5 \in A$ is false.
The set $\{3\}$ has its only member in $A$ , so $\{3\} \subseteq A$ is true.

$3 \in A$ ✓ · $5 \in A$ ✗ · $\{3\} \subseteq A$ ✓

Classic trap

Trap:

\{0\}

is NOT empty — it holds one element.

\{x \in \mathbb{R} : x^2+1 = 0\}

is empty because no real

x

works.

Lesson 2 of 16

Set Representation

Learn the idea

Every set can be written three ways. **Roster form** lists elements:

A = \{2, 4, 6, 8\}

. **Set-builder form** gives a rule:

A = \{x \in \mathbb{Z} : x \text{ is even and } 2 \le x \le 8\}

. **Descriptive form** uses words.

The exam loves converting between forms — read the rule, list the elements, and check endpoints.

Key rules to memorize

Number sets: $\mathbb{N}$ naturals, $\mathbb{Z}$ integers, $\mathbb{Q}$ rationals, $\mathbb{R}$ reals, with $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$ .
Watch whether $\mathbb{N}$ starts at 0 or 1 — follow the question's convention.

Worked example

Write $A = \{x \in \mathbb{Z} : x \text{ is even and } 2 \le x \le 8\}$ in roster form.

Even integers from $2$ to $8$ : test $2, 3, 4, 5, 6, 7, 8$ .
Keep the evens, endpoints included.

$A = \{2, 4, 6, 8\}$

Classic trap

Trap: set-builder conditions like

2 \le x \le 8

include both endpoints — don't drop them when listing.

Lesson 3 of 16

Subsets and Power Sets

Learn the idea

A \subseteq B

means every element of

A

is in

B

. Every set is a subset of itself, and

\varnothing \subseteq A

always. A **proper** subset excludes equality:

A \subseteq B

and

A \ne B

Counting is mechanical: an

n

-element set has

2^n

subsets and

2^n - 1

proper subsets, because each element is either in or out.

Key rules to memorize

Subsets: $\boxed{2^n}$ . Proper subsets: $\boxed{2^n - 1}$ .
Power set $\mathcal{P}(A)$ collects all subsets: if $A = \{1, 2\}$ then $\mathcal{P}(A) = \{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$ , so $|\mathcal{P}(A)| = 2^{|A|}$ .
Subsets containing one fixed element: fix it in, choose freely from the rest → $2^{n-1}$ .
Exactly- $k$ subsets: choose which $k$ → $C(n,k)$ ; e.g. an 8-element set has $C(8,3) = 56$ three-element subsets.

Worked example

Set $A$ has $4$ elements. How many subsets and proper subsets?

Each element is in or out: $2^4 = 16$ subsets.
Proper drops the set itself: $16 - 1 = 15$ .

$16$ subsets · $15$ proper

Classic trap

Trap:

\{1, 3\} \not\subseteq \{1, 2\}

— one outsider breaks a subset. Always check every element.

Lesson 4 of 16

Set Operations

Learn the idea

Three operations do all the work. **Union**

A \cup B

collects elements in either set (OR). **Intersection**

A \cap B

keeps only common elements (AND). **Difference**

A - B

keeps elements of

A

outside

B

With

A = \{1, 2, 3, 4\}

and

B = \{3, 4, 5, 6\}

A \cup B = \{1, 2, 3, 4, 5, 6\}

A \cap B = \{3, 4\}

A - B = \{1, 2\}

Key rules to memorize

Complement $A^c = U - A$ needs the universal set $U$ : with $U = \{1, \ldots, 6\}$ and $A = \{1, 2, 3\}$ , $A^c = \{4, 5, 6\}$ .
Sizes follow: $|A - B| = |A| - |A \cap B|$ and $|A^c| = |U| - |A|$ .
Cardinality: $|A|$ counts elements; Cartesian product $|A \times B| = |A| \cdot |B|$ .

Worked example

With $A = \{1, 2, 3, 4\}$ , $B = \{3, 4, 5, 6\}$ , find $A \cup B$ , $A \cap B$ , $A - B$ .

Union combines once: $\{1, 2, 3, 4, 5, 6\}$ .
Intersection keeps commons: $\{3, 4\}$ .
Difference drops $B$ 's items from $A$ : $\{1, 2\}$ .

$\cup = \{1,2,3,4,5,6\}$ · $\cap = \{3,4\}$ · $- = \{1,2\}$

Classic trap

Trap:

A - B \ne B - A

in general, and you cannot find a complement without knowing

U

Lesson 5 of 16

Set Laws

Learn the idea

Union and intersection are commutative and associative, and they distribute over each other:

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Identity and domination laws handle the edge cases:

A \cup \varnothing = A

A \cap U = A

A \cup U = U

A \cap \varnothing = \varnothing

Key rules to memorize

Complement laws: $A \cup A^c = U$ , $A \cap A^c = \varnothing$ , $(A^c)^c = A$ .
Commutative: $A \cup B = B \cup A$ , $A \cap B = B \cap A$ .
Associative: $(A \cup B) \cup C = A \cup (B \cup C)$ , same for $\cap$ .
Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
Identity: $A \cup \varnothing = A$ , $A \cap U = A$ . Domination: $A \cup U = U$ , $A \cap \varnothing = \varnothing$ .
Idempotent: $A \cup A = A$ , $A \cap A = A$ .
De Morgan (boxed): $(A \cup B)^c = A^c \cap B^c$ , $(A \cap B)^c = A^c \cup B^c$ — NOT-OR becomes AND-NOT.
If $A \subseteq B$ then $A \cap B = A$ and $A \cup B = B$ — instant simplifications.

Worked example

Simplify $(A \cup B)^c$ .

De Morgan: complement flips $\cup$ to $\cap$ .
Push the complement inside both sets.

$A^c \cap B^c$

Classic trap

Trap: distribution looks like algebra but isn't:

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

, with union outside AND inside.

Lesson 6 of 16

Venn Diagrams

Learn the idea

Translate wording before counting.

A

B

→

A \cup B

A

and

B

→

A \cap B

A

but not

B

→

A - B

. Neither →

(A \cup B)^c

Fill the diagram inside-out: both/all-three first, then only-regions, then the outside.

Key rules to memorize

Only A = $A - B$ ; exactly one of A, B = A-only + B-only.
Exactly two of A, B, C = pairwise-only regions; all three = triple intersection.
Neither A nor B = total − $|A \cup B|$ .
“At least one” → union; “both” → intersection.
“Or” is inclusive unless the question says “but not both”.

Worked example

40 students: 25 like math, 20 like physics, 12 both. How many like neither?

At least one: $25 + 20 - 12 = 33$ .
Neither: $40 - 33 = 7$ .

$7$ students

Classic trap

Trap: “only A” is

A - B

, not

A

— subtract the overlap before answering.

Lesson 7 of 16

Inclusion-Exclusion

Learn the idea

Adding

|A| + |B|

counts the overlap twice, so subtract it once:

|A \cup B| = |A| + |B| - |A \cap B|

Three sets extend the same idea — add singles, subtract pairs, add back the triple:

|A \cup B \cup C| = |A|+|B|+|C| - |A\cap B|-|B\cap C|-|C\cap A| + |A\cap B\cap C|

Key rules to memorize

A-only region: $|A| - |A \cap B| - |A \cap C| + |A \cap B \cap C|$ .
Disjoint sets: overlap is 0, so $|A \cup B| = |A| + |B|$ .
Max union $= |A| + |B|$ (disjoint); min union $= \max(|A|, |B|)$ (one inside the other).

Worked example

$|A| = 30$ , $|B| = 25$ , $|A \cap B| = 10$ . Find $|A \cup B|$ .

Adding counts the overlap twice.
Subtract it once: $30 + 25 - 10 = 45$ .

$45$

Classic trap

Trap: forgetting to add back the triple intersection — pairs subtract it three times but it was added three times, so net needs +1.

Lesson 8 of 16

Cartesian Products

Learn the idea

A \times B

holds ordered pairs

(a, b)

with

a \in A, b \in B

. Counting is multiplication:

|A \times B| = |A| \cdot |B|

Order matters:

(a, b) \ne (b, a)

unless the coordinates are equal.

Key rules to memorize

Example: $|A| = 4, |B| = 3$ gives $4 \times 3 = 12$ ordered pairs.

Worked example

$|A| = 4$ , $|B| = 3$ . How many pairs in $A \times B$ ?

Each of the $4$ A-items pairs with each of the $3$ B-items.
Multiply: $4 \times 3 = 12$ .

$12$ ordered pairs

Classic trap

Trap:

|A \times B|

multiplies (

mn

), it is not

m + n

— don't confuse pairs with unions.

Lesson 9 of 16

Inequality Basics

Learn the idea

Solve like an equation — isolate

x

— with one extra rule: multiplying or dividing by a **negative** flips the sign:

-2x > 6 \Rightarrow x < -3

Adding or subtracting never flips anything.

Key rules to memorize

Interval notation: $x > 2 \Rightarrow (2, \infty)$ , $x \le 5 \Rightarrow (-\infty, 5]$ . Infinity always takes a round bracket.
$2 < x + 1 \le 7$ is an AND: subtract 1 everywhere → $1 < x \le 6$ .

Worked example

Solve $-2x > 6$ .

Divide by $-2$ — negative, so flip the sign.
Get $x < -3$ .

$x < -3$

Classic trap

Trap: decide AND vs OR first.

x > 2

and

x < 7

combine to

2 < x < 7

; “or” stays two separate intervals.

Lesson 10 of 16

Compound Inequalities

Learn the idea

AND means both must hold (one merged interval); OR means at least one holds (separate intervals). Example:

-4 \le 2x + 2 < 8

→ subtract 2, divide by 2 →

-3 \le x < 3

Word codes: at least

\ge

, at most

\le

, no more than

\le

, no less than

\ge

Key rules to memorize

“At least 10” is $x \ge 10$ , not $x > 10$ . “At most 10” is $x \le 10$ .

Worked example

Solve $-4 \le 2x + 2 < 8$ .

Subtract $2$ everywhere: $-6 \le 2x < 6$ .
Divide by $2$ (positive, no flip): $-3 \le x < 3$ .

$-3 \le x < 3$

Classic trap

Trap: never auto-merge two inequalities — check whether the wording is AND or OR before combining.

Lesson 11 of 16

Absolute Value

Learn the idea

|x|

is distance from zero, never negative.

|x| = a

(with

a > 0

) splits:

x = a

x = -a

; shifted,

|x - c| = a

gives

x = c \pm a

Inequalities follow one memory rule: **< means AND, > means OR**.

|x| < a \iff -a < x < a

, while

|x| > a \iff x < -a

x > a

Key rules to memorize

Equations first: $|x| = a \Rightarrow x = \pm a$ ; shifted $|x - c| = a \Rightarrow x = c \pm a$ (distance $a$ from $c$ ).
$|x - 3| \le 5 \iff -5 \le x - 3 \le 5 \iff -2 \le x \le 8$ .
Two absolute values ( $|2x-3| < |x+1|$ ): square both sides, then run a sign chart — here $\frac23 < x < 4$ .
Sums like $|x-2| + |x+2| \le 6$ : split at the critical points $-2, 2$ into three regions, drop each $|·|$ with the correct sign per region, and solve — here $-3 \le x \le 3$ . Shortcut: read it as distance — points whose distances to $2$ and $-2$ total at most 6.

Worked example

Solve $|x - 3| \le 5$ .

$\le$ means AND: $-5 \le x - 3 \le 5$ .
Add $3$ : $-2 \le x \le 8$ .

$-2 \le x \le 8$

Classic trap

Trap: treating

|x| > a

as an AND (a single interval) — it is always two rays.

Lesson 12 of 16

Quadratic Inequalities

Learn the idea

Recipe: move everything to one side, factor, find zeros, sign-chart the intervals, then include/exclude endpoints by

\le/\ge

</>

For

(x-a)(x-b)

with

a < b

and positive leading coefficient: outside the roots is positive, between is negative.

Key rules to memorize

$x^2 - 5x + 6 \le 0 \Rightarrow (x-2)(x-3) \le 0 \Rightarrow 2 \le x \le 3$ .
$x^2 - 6x + 9 = (x-3)^2 \ge 0$ always, so $(x-3)^2 < 0$ has no real solution.

Worked example

Solve $x^2 - 5x + 6 < 0$ .

Factor: $(x-2)(x-3) < 0$ , roots $2, 3$ .
Upward parabola is negative between the roots.

$2 < x < 3$

Classic trap

Trap: including a root under a strict

</>

sign, or forgetting that a repeated (even-multiplicity) root doesn't change sign.

Lesson 13 of 16

Polynomial Inequalities

Learn the idea

Same sign-chart engine, more factors: order all real zeros, test one point per interval, keep the intervals with the wanted sign.

Multiplicity decides: odd multiplicity flips the sign at the root, even multiplicity touches and keeps it.

Key rules to memorize

$(x+1)(x-2)(x-4) > 0 \Rightarrow (-1, 2) \cup (4, \infty)$ .
$(x-1)^2(x+2) < 0$ : $(x-1)^2 \ge 0$ (zero only at $x = 1$ , excluded), so need $x + 2 < 0 \Rightarrow x < -2$ .

Worked example

Solve $(x+1)(x-2)(x-4) > 0$ .

Zeros in order: $-1, 2, 4$ .
Rightmost interval positive, signs alternate: $(-1,2)$ and $(4,\infty)$ .

$(-1,2) \cup (4,\infty)$

Classic trap

Trap: expanding a factored polynomial — the factored form IS the sign chart input; expanding wastes time and invites errors.

Lesson 14 of 16

Rational Inequalities

Learn the idea

Collect numerator zeros AND denominator zeros, order them, sign-chart, then select. The golden rule: **denominator zeros are always excluded**, even under

\ge/\le

Example:

\frac{x-1}{x+3} > 0

→ critical

x = 1, x = -3

(excluded) →

x < -3

x > 1

Key rules to memorize

$\frac{x-3}{x+2} \ge 0 \Rightarrow (-\infty, -2) \cup [3, \infty)$ : numerator zero $3$ included, denominator zero $-2$ excluded.
$\frac{1}{(x-2)(x+3)} \le 0 \Rightarrow -3 < x < 2$ : numerator never zero, both critical points excluded.

Worked example

Solve $\frac{x-1}{x+3} > 0$ .

Critical points: numerator zero $x = 1$ , denominator zero $x = -3$ (always excluded).
Same sign holds outside: $x < -3$ or $x > 1$ .

$x < -3$ or $x > 1$

Classic trap

Trap: cross-multiplying by a denominator of unknown sign, or simplifying away a factor and forgetting its zero stays excluded.

Lesson 15 of 16

Word Problems

Learn the idea

Translate first, solve second. “At least 40” →

\ge 40

; “no more than 20 and greater than 5” →

(5, 20]

Two-variable

y > 2x + 1

shades **above** the boundary; strict

</>

uses a dashed (excluded) boundary,

\le/\ge

a solid one.

Key rules to memorize

Rectangle width $x$ , length $x+3$ , area at least 40: $x(x+3) \ge 40$ with $x > 0$ .
“5 units from 2, at least”: $|x - 2| \ge 5$ .
Full codebook: greater than $>$ , less than $<$ , at least $\ge$ , at most $\le$ , no more than $\le$ , no less than $\ge$ , minimum $\ge$ , maximum $\le$ , more than $>$ , fewer than $<$ .

Worked example

Width $x$ , length $x+3$ , area at least $40$ , $x > 0$ . Set up the inequality.

Area = width × length = $x(x+3)$ .
“At least 40” → $\ge 40$ .

$x(x+3) \ge 40$

Classic trap

Trap: “at least” (

\ge

) vs “more than” (

>

) — one boundary point decides the option.

Lesson 16 of 16

Two-Variable Inequalities

Learn the idea

A linear inequality in two variables like

y > 2x + 1

describes a **region** of the coordinate plane, cut by its boundary line

y = 2x + 1

Shading rule:

y > f(x)

shades **above** the boundary;

y < f(x)

shades **below** it. Test with

(0,0)

when unsure — if it satisfies the inequality, its side is shaded.

Key rules to memorize

Strict $</>$ → dashed boundary (excluded); $\le/\ge$ → solid boundary (included).
Example: $y > 3x - 2$ is the region strictly above the line $y = 3x - 2$ .
Vertical/horizontal cases work the same way: $x > 1$ is everything right of the line $x = 1$ .

Worked example

Where is the solution of $y > 2x + 1$ ? Is the boundary included?

$y >$ means the region above the line $y = 2x + 1$ .
Strict $>$ → dashed boundary (excluded).

Above the line, boundary excluded

Classic trap

Trap: shading the wrong side, or drawing a solid line for a strict inequality — always test one point.

Master formula sheet

Counting

Subsets of an $n$ -element set: $\boxed{2^n}$
Proper subsets: $\boxed{2^n-1}$
Power set size: $|\mathcal{P}(A)| = 2^{|A|}$
Subsets containing a fixed element: $2^{n-1}$
Cartesian product: $|A \times B| = |A| \cdot |B|$

Operations & laws

Complement: $A^c = U - A$
De Morgan: $(A \cup B)^c = A^c \cap B^c$ , $(A \cap B)^c = A^c \cup B^c$
Two-set union: $|A \cup B| = |A| + |B| - |A \cap B|$
Three-set union: add singles, subtract pairs, add back the triple
If $A \subseteq B$ : $A \cap B = A$ , $A \cup B = B$

Absolute value

$|x| < a \iff -a < x < a$ (AND)
$|x| \le a \iff -a \le x \le a$
$|x| > a \iff x < -a \text{ or } x > a$ (OR)
$|x| \ge a \iff x \le -a \text{ or } x \ge a$

Inequality engine

Negative multiply/divide flips: $< \leftrightarrow >$ , $\le \leftrightarrow \ge$
Quadratic $(x-a)(x-b)$ , $a<b$ : $>0$ outside, $<0$ between
Odd multiplicity flips sign; even multiplicity keeps it
Rational: numerator zeros + denominator zeros, sign chart, denominator always excluded

Traps & recognition

15 classic traps

Confusing $\in$ (element) with $\subseteq$ (subset).
Forgetting $\varnothing$ is a subset of every set.
Counting repeated elements twice.
Skipping the subtracted intersection in union counting.
Dropping the added-back triple intersection for three sets.
Treating $A-B$ and $B-A$ as the same.
Finding a complement without fixing the universal set $U$ .
Not flipping the sign after negative division/multiplication.
Reading $|x|>a$ as AND instead of OR.
Including denominator zeros (never allowed).
Including roots under strict $</>$ signs.
Forgetting even-multiplicity roots keep their sign.
Drawing solid boundaries for strict inequalities.
Mixing up “at least” ( $\ge$ ) with “more than” ( $>$ .
Expanding factored polynomials instead of sign-charting them.

Fast problem recognition

“How many subsets?”
Use $2^n$ (proper: $2^n-1$ ).
“Common / both”
Think $A \cap B$ .
“Either / or / at least one”
Think $A \cup B$ + inclusion–exclusion.
“Not in / neither”
Think complement / difference.
Linear inequality
Isolate $x$ ; watch negative division.
Absolute value
< → AND interval; > → OR rays (or square both sides).
Quadratic / polynomial
Roots + sign chart; check multiplicity.
Rational
Numerator + denominator critical points; exclude denominator zeros.

Practice — 300 problems

0/0 correct

Q1basic · membership
If $A=\{1,2,3,4\}$ , which statement is true?
Q2basic · cardinality
If $A=\{a,b,c\}$ , then $|A|=$
Q3basic · empty-set
Which is an empty set?
Q4basic · subset
How many subsets does a 5-element set have?
Q5basic · subset
How many proper subsets does a 5-element set have?
Q6basic · intersection
If $A=\{1,2,3\}$ and $B=\{3,4,5\}$ , then $A\cap B=$
Q7basic · union
For the same A and B, $A\cup B=$
Q8basic · difference
If $A=\{1,2,3,4\}$ and $B=\{2,4\}$ , then $A-B=$

Previous CSCA exam questions

Real questions from the Dec 2025, Jan 2026 and Apr 2026 papers that test this chapter — solved with the thinking shown. Answers solved by the app (the source papers publish no official key); items with source errors are marked and still teach the correct math.

14 exam questions · 0/0 correct

Dec 2025 · Q1sets
If sets $A = \{1,2\}$ , $B = \{1,2,3\}$ , then
Dec 2025 · Q2sets
If sets $A = (-1,3)$ , $B = [2,4]$ , then $A \cap B =$
Dec 2025 · Q3inequalities
The solution set of $x^2 - 5x + 4 < 0$ is
Dec 2025 · Q12inequalities
The solution set of $\frac{x-1}{x-2} \le 1$ is
Jan 2026 · Q1sets
$A = \{6,7,8,9,10\}$ : which is correct?
Jan 2026 · Q2sets
$A = \{x | -2 \le x \le 2\}$ , $B = \{x | x > 0\}$ : $A \cup B =$
Jan 2026 · Q3inequalities
$x^2 - x - 2 > 0$ : solution set is
Jan 2026 · Q11inequalities
$\frac{2x+1}{x-2} \le 0$ : solution set is

Exam checklist

0/14 checked

Sets

Inequalities

One-minute memory sheet

Union = OR, Intersection = AND
Subsets $= 2^n$ , proper $= 2^n-1$
$|A \cup B| = |A| + |B| - |A \cap B|$
Negative multiply/divide = flip the sign
$|x| < a \Rightarrow$ AND, $|x| > a \Rightarrow$ OR
Quadratic/polynomial = roots + sign chart
Rational = numerator + denominator zeros + sign chart
Denominator zero = ALWAYS excluded

Related guides

30-day study plan → fit this chapter into a week-by-week system
75-second pacing rule → hold 75s/question once the content clicks
Top 10 mistakes → the error patterns that cost the most points

Last reviewed Sep 2026 · GetCSCA editors

Timed practice →Back to syllabus →