Venn Diagrams; Cubes and Dice; Mirror and Water Images; Paper Folding

Venn Diagrams

20.1 Set Notation

n(A) = Number of elements in A
n(A∪B) = n(A) + n(B) - n(A∩B)
n(A∩B) = Elements in both A and B
n(A') = Elements NOT in A = Total - n(A)

20.2 Three Sets Formula

n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)

20.3 Diagram Types

All A are B:

   ┌─────────────B─────────────┐
   │      ┌──A──┐              │
   │      └─────┘              │
   └───────────────────────────┘

Some A are B:

  ┌──A──┐ ┌──B──┐
  │  ┌──┼─┼──┐  │
  │  │  │ │  │  │
  └──┼──┘ └──┼──┘
     └────────┘

No A is B:

  ┌──A──┐    ┌──B──┐
  │     │    │     │
  └─────┘    └─────┘

20.4 Questions Format

"In a group of 100, 60 like cricket, 50 like football and 30 like both. How many like at least one?"

n(C∪F) = n(C) + n(F) - n(C∩F) = 60 + 50 - 30 = 80
Neither = 100 - 80 = 20

Cubes and Dice

21.1 Cube Properties

A cube has 6 faces, 12 edges, 8 vertices (corners)
Each face is a square
When a cube is cut n times in each dimension → (n+1)³ pieces

21.2 Painting of Cubes

When a cube is painted on all faces and cut into (n×n×n) small cubes:

Position	Faces Painted	Count
Corner cubes	3 faces	8 (always)
Edge cubes (not corners)	2 faces	12(n-2)
Face cubes (not edges)	1 face	6(n-2)²
Interior cubes	0 faces	(n-2)³

Example: Cube cut into 4×4×4 = 64 small cubes (n=4):

3 faces: 8
2 faces: 12(2) = 24
1 face: 6(4) = 24
0 faces: (2)³ = 8
Total: 8 + 24 + 24 + 8 = 64 ✓

21.3 Dice Problems

Standard Die:

Opposite faces sum to 7: (1,6), (2,5), (3,4)
When 1 is on top, 6 is at bottom
When 2 faces you, 5 is behind

Open Dice / Unfolded Dice:

Given flat pattern, determine which faces are opposite

The face directly across in unfolded form = NOT opposite

Corners never share the same die face

Two Dice Questions:

Given views of the same die from two positions, determine hidden faces
One common face in two views → use rotation to find others

21.4 Determining Opposite Faces (from multiple views)

Given three views of same die:

Identify the common face
Find which face changes in each rotation
Use elimination to identify all opposite pairs

Mirror and Water Images

22.1 Mirror Image

Mirror placed on the RIGHT side of an object → Left-right reversal

A → mirror image looks like a reversed A

Rules:

Vertical (left-right) mirror → left becomes right, right becomes left

Horizontal (top-bottom) mirror → top becomes bottom, bottom becomes top

A clock in front of a vertical mirror: 3 o'clock appears as 9 o'clock

22.2 Mirror Image of Clock

Formula: Mirror time = 11:60 - Given time

Mirror time = 11 hours 60 minutes - clock time

Example: Clock shows 3:25 → Mirror image = 11:60 - 3:25 = 8:35

Alternative method: Subtract from 12:00

12:00 - 3:25 = 8:35 ✓

22.3 Water Image

Water image = Vertical flip (upside down) of the object

Top becomes bottom; bottom becomes top
Left-right stays the SAME

Water image of letter:

A → ∀ (inverted A)
B → looks like B reflected over horizontal axis

22.4 Mirror Image of Letters

Letters that look the same in mirror: A, H, I, M, O, T, U, V, W, X, Y Letters that look the same in water image: B, C, D, E, H, I, K, O, X

Paper Folding

23.1 Concept

A paper is folded and then a hole is punched. Determine where holes appear when paper is unfolded.

23.2 Method

Track each fold direction (left-right or top-bottom)
Unfold in reverse order
Each hole creates a mirror image of itself when unfolded

Example:

Fold left half over right → punch hole in center → 2 holes when unfolded (symmetric about fold line)

23.3 Types of Folds

Left to Right fold: mirror along vertical axis
Top to Bottom fold: mirror along horizontal axis
Diagonal fold: mirror along diagonal axis