Sets & Logic | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Understand set notation and represent sets using Venn diagramsFahami notasi set dan wakilkan set menggunakan gambar rajah Venn

Perform set operations: union, intersection, complementLakukan operasi set: kesatuan, persilangan, pelengkap

Use Venn diagrams to solve problems involving 2 or 3 setsGunakan gambar rajah Venn untuk selesaikan masalah 2 atau 3 set

Apply set concepts to solve word problemsGunakan konsep set untuk selesaikan masalah perkataan

📖 Key Concepts📖 Konsep Utama

Set Notation (Form 4)Notasi Set (Tingkatan 4)

A set is a collection of objects. Elements are listed or described.Set ialah koleksi objek. Unsur disenaraikan atau diperikan.

∈ = element of | ∉ = not an element of
n(A) = number of elements
∅ = empty set | ξ = universal set

Set Operations (Form 4)Operasi Set (Tingkatan 4)

Union, intersection and complement combine sets in different ways.Kesatuan, persilangan dan pelengkap menggabungkan set dengan cara berbeza.

Union (A∪B)Kesatuan (A∪B)

Intersection (A∩B)Persilangan (A∩B)

Complement (A')Pelengkap (A')

A ∪ B = union (all elements in A or B)
A ∩ B = intersection (common elements)
A' = complement (not in A)

Venn Diagrams (Form 4)Gambar Rajah Venn (Tingkatan 4)

Visual representation of sets and their relationships.Perwakilan visual set dan hubungannya.

2-Set VennVenn 2 Set

3-Set VennVenn 3 Set

Overlap = intersection
Outside = complement
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

De Morgan's Laws (Form 5)Hukum De Morgan (Tingkatan 5)

Rules for complement of union and intersection.Peraturan untuk pelengkap kesatuan dan persilangan.

(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'

✏️ Worked Examples✏️ Contoh Penyelesaian

If n(A) = 12, n(B) = 8, n(A ∩ B) = 3, find n(A ∪ B)Jika n(A) = 12, n(B) = 8, n(A ∩ B) = 3, cari n(A ∪ B)

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

n(A ∪ B) = 12 + 8 - 3 = 17

n(A ∪ B) = 17

If A = {1,2,3,4}, B = {3,4,5,6}, find A ∩ BJika A = {1,2,3,4}, B = {3,4,5,6}, cari A ∩ B

A ∩ B = elements common to both sets

Common elements: 3 and 4

A ∩ B = {3, 4}

ξ = {1,2,...,10}, A = {1,3,5,7}, find A'ξ = {1,2,...,10}, A = {1,3,5,7}, cari A'

A' = ξ - A (elements in universal set but not in A)

A' = {2, 4, 6, 8, 10}

Verify De Morgan's Law: (A ∪ B)' = A' ∩ B' for A = {1,2,3}, B = {2,3,4}, ξ = {1,2,3,4,5}Sahkan Hukum De Morgan: (A ∪ B)' = A' ∩ B' untuk A = {1,2,3}, B = {2,3,4}, ξ = {1,2,3,4,5}

A ∪ B = {1,2,3,4}, so (A ∪ B)' = {5}

A' = {4,5}, B' = {1,5}

A' ∩ B' = {5} = (A ∪ B)' ✓

(A ∪ B)' = {5} = A' ∩ B' ✓

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Forgetting to subtract the intersection in n(A ∪ B)Lupa menolak persilangan dalam n(A ∪ B)

n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Don't just add!n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Jangan hanya tambah!

❌ Confusing union (∪) with intersection (∩)Mengelirukan kesatuan (∪) dengan persilangan (∩)

Union = ALL elements (or). Intersection = ONLY common elements (and).Kesatuan = SEMUA unsur (atau). Persilangan = HANYA unsur sepunya (dan).

❌ Misreading Venn diagram regionsMembaca kawasan gambar rajah Venn dengan salah

Label every region carefully. The overlap belongs to BOTH sets. The outside is the complement.Label setiap kawasan dengan teliti. Tindihan milik KEDUA-DUA set. Luar ialah pelengkap.

📝 SPM Practice📝 Amalan SPM

Paper 1 - Multiple ChoiceKertas 1 - Pilihan Berganda

If n(A) = 12, n(B) = 8, n(A ∩ B) = 3, find n(A ∪ B)Jika n(A) = 12, n(B) = 8, n(A ∩ B) = 3, cari n(A ∪ B)

(A ∪ B)' is equal to:(A ∪ B)' sama dengan:

A' ∪ B'

A' ∩ B'

A ∩ B

A ∪ B

If A = {1,2,3} and B = {2,3,4}, find A ∩ BJika A = {1,2,3} dan B = {2,3,4}, cari A ∩ B

{1,2,3,4}

{2,3}

{1,4}

{1,2,3}

In a Venn diagram, the region outside both circles represents:Dalam gambar rajah Venn, kawasan di luar kedua-dua bulatan mewakili:

A ∪ B

A ∩ B

(A ∪ B)'

∅

If ξ = {1,2,...,10}, A = {1,3,5,7}, find A'Jika ξ = {1,2,...,10}, A = {1,3,5,7}, cari A'

{2,4,6,8,10}

{1,3,5,7}

{2,4,6,8}

{1,2,...,10}

Paper 2 - Structured QuestionsKertas 2 - Soalan Berstruktur

5 marks5 markah

In a class of 40 students, 25 play football, 20 play basketball, and 10 play both.
(a) Draw a Venn diagram. [2 marks]
(b) Find the number who play neither sport. [3 marks]Dalam kelas 40 pelajar, 25 bermain bola sepak, 20 bermain bola keranjang, dan 10 bermain kedua-duanya.
(a) Lukis gambar rajah Venn. [2 markah]
(b) Cari bilangan yang tidak bermain kedua-dua sukan. [3 markah]

Solution (a)Penyelesaian (a)

Football only: 25 - 10 = 15

Basketball only: 20 - 10 = 10

Both: 10 (overlap region)

Venn diagram with 15, 10, 10 in regions [2 marks]

Solution (b)Penyelesaian (b)

Total in diagram: 15 + 10 + 10 = 35

Neither: 40 - 35 = 5

5 students play neither sport [3 marks]

6 marks6 markah

Given ξ = {x : 1 ≤ x ≤ 12, x is an integer}, A = {2,4,6,8,10,12}, B = {3,6,9,12}.
(a) List the elements of A ∩ B. [2 marks]
(b) Find (A ∪ B)'. [2 marks]
(c) Verify that (A ∪ B)' = A' ∩ B'. [2 marks]Diberi ξ = {x : 1 ≤ x ≤ 12, x ialah integer}, A = {2,4,6,8,10,12}, B = {3,6,9,12}.
(a) Senaraikan unsur A ∩ B. [2 markah]
(b) Cari (A ∪ B)'. [2 markah]
(c) Sahkan bahawa (A ∪ B)' = A' ∩ B'. [2 markah]

Solution (a)Penyelesaian (a)

A ∩ B = common elements of A and B

A ∩ B = {6, 12}

A ∩ B = {6, 12} [2 marks]

Solution (b)Penyelesaian (b)

A ∪ B = {2,3,4,6,8,9,10,12}

(A ∪ B)' = ξ - (A ∪ B) = {1,5,7,11}

(A ∪ B)' = {1, 5, 7, 11} [2 marks]

Solution (c)Penyelesaian (c)

A' = {1,3,5,7,9,11}, B' = {1,2,4,5,7,8,10,11}

A' ∩ B' = {1,5,7,11}

(A ∪ B)' = A' ∩ B' ✓