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CLASS 8 • SETS / CYBER MATH

OPERATE
ON SETS.

See how sets combine, overlap, separate and sit inside a universal set. Build your own sets, test the operations, and use cardinal numbers to check whether the result makes sense.

CURRENT MISSION
Decode the overlap.
A = {2, 4, 6, 8, 10} and B = {6, 8, 12}. Find A ∩ B.
LEARNING CORE
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01 // SET CONSOLE

Build A and B inside U.

Tap a number to switch its membership. Select A, B, or both as the target. The operation panel updates instantly.

CYAN = A PINK = B VIOLET = A ∩ B
UNIVERSAL SET U = {1, 2, ..., 12}
Editing A. Tap elements to add or remove them.
U
A
B
A ∩ B
A
∅
B
∅
OPERATION SNAPSHOT
A ∪ B = ∅
02 // UNION

Join without duplication.

The union keeps every element that appears in A, in B, or in both. If an element is present in both, it is written only once.

{2, 4, 6, 8} ∪ {6, 8, 10, 12}
Think: scan both sets, collect everything, and erase repeated copies.
= {2, 4, 6, 8, 10, 12}

Union in plain language

A ONLY
2, 4
from A
BOTH
6, 8
shared
B ONLY
10, 12
from B
SYMBOL A ∪ B
KEY IDEA at least one set
03 // INTERSECTION

Keep only the overlap.

The intersection contains elements that belong to both sets. It is the shared zone.

{2, 4, 6, 8} ∩ {6, 8, 10, 12}
Think: look for values you can point to in A and also point to in B.
= {6, 8}

Intersection test

6
IN
in A and B
10
OUT
only in B
4
OUT
only in A, not in both
8
IN
in both
Important: for intersection, “or” is not enough. The element must satisfy both memberships.
04 // DIFFERENCE

Direction matters.

A − B means elements in A that are not in B. B − A is a different set, because the starting set has changed.

{2, 4, 6, 8} − {6, 8, 10, 12}
= {2, 4}
Rule: start with the first set. Remove every element that appears in the second set.

A − B keeps the A-only elements.
05 // COMPLEMENT

Everything in U that A does not contain.

The complement is always relative to a chosen universal set. That context matters.

U = {1, 2, 3, ..., 12}
A = {2, 4, 6, 8}
So A′: scan U and keep the elements missing from A.
A′ = {1, 3, 5, 7, 9, 10, 11, 12}

Complement checkpoint

A ∪ A′
U
everything is covered
A ∩ A′
∅
nothing overlaps
FINITE U
n(A)+n(A′)
= n(U)
06 // SET RELATIONS

Overlap. Disjoint. Subset.

Use the intersection to diagnose the relationship between two sets, and remember that an empty intersection is the signal for disjointness.

Overlapping

A ∩ B ≠ ∅

At least one element is shared.

Disjoint

A ∩ B = ∅

There is no common element.

Subset

A ⊆ B

Every element of A is also in B.

Law scanner

A ∪ B = B ∪ A
Swap the order and the union stays the same. The same is true for intersection.
07 // CARDINALITY ENGINE

Count without recounting.

Cardinal number means the number of distinct elements in a finite set. The overlap prevents double-counting.

The intersection cannot be larger than either set. The controls are automatically limited so the numbers stay sensible.
UNION SIZE
10
A ONLY
4
B ONLY
3
7 + 6 − 3 = 10

Three cardinal formulas worth owning

n(A ∪ B)
n(A)+n(B)−n(A∩B)
n(A−B)
n(A)−n(A∩B)
n(B−A)
n(B)−n(A∩B)
08 // MEMORY DECK

Fast recall, no guessing.

Flip the cards, explain the idea in your own words, then reshuffle for a new order.

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ROOT//8 DEEP CARD
Why can't we calculate A′ without knowing the universal set U?
Tap to reveal.
REVEAL
Because the complement contains elements of U that are not in A. Change U and the complement can change too.
09 // PRACTICE LAB

Calculate, then verify.

Choose a mission type. Each new problem is fresh. Three attempts protect the learner from random guessing.

Find A ∪ B when A = {1, 3, 5, 7} and B = {3, 4, 7, 8}.
Hint: combine the elements, but do not repeat shared values.
10 // WORD OPS

Translate words into set operations.

These are built around school-style situations: clubs, sports, languages, numbers, attendance and survey data.

11 // SLIDE DECK

Teach the chapter like a briefing.

Use the arrows in class, or let a learner step through the ideas independently.

ROOT//8 BRIEFING 01

Union

A ∪ B

Keep everything that belongs to A, B, or both. Repeated elements appear once.

BRIEFING 02

Intersection

A ∩ B

Keep only the shared elements.

BRIEFING 03

Difference

A − B

Start with A and remove the elements that also occur in B.

BRIEFING 04

Complement

A′

Everything in U that is outside A.

BRIEFING 05

De Morgan

(A ∪ B)′ = A′ ∩ B′

Complementing a union turns the connection into an intersection of complements.

BRIEFING 06

Cardinality

n(A∪B)=n(A)+n(B)−n(A∩B)

Subtract the overlap because it was counted twice.

12 // SYSTEM CHECK

Three attempts. Then reveal.

Every new set creates fresh Class 8 set-operation questions. The learner sees the answer after a correct response or three misses.

Choose an answer. Three attempts are allowed for each question.
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