SETS

🧠 SETS — GEN Z REVIEWER (DISCRETE MATH)

1. What is a Set?

👉 Foundation ito, so lock this in first.

A set is:

• A well-defined

• Unordered collection of objects

• Objects are called elements or members

• Usually written using capital letters

📌 Unordered = walang pakialam ang order
{1,2,3} = {3,2,1}

✅ Well-defined vs ❌ Not well-defined

🧠 Mnemonic:
Set = sure, clear, walang opinion

2. Set Notation

🔹 Roster / Listing Method

• Nililista lahat ng elements

Example:
S = {0,1,2,3,4,5,6,7,8,9}

🔹 Descriptive / Set-Builder Method

• Dinidescribe kung ano ang elements

Example:
S = {whole numbers less than 10}

🧠 Tip:
Roster = lista
Set-builder = description

3. Universal Set (U)

• Set na naglalaman ng lahat ng elements na pinag-uusapan

• Usually symbol: U

Example:
U = {0,1,2,3,4,5,6,7,8,9}

🧠 Think of it as:
“Big container” ng lahat

4. Element Notation

Example:

• If E = {even numbers}
  2 ∈ E
  3 ∉ E

5. Finite vs Infinite Sets

🔹 Finite Set

• Countable ang elements

Example:

• {whole numbers less than 5} → 5 elements

• Alphabet → 26 letters

🔹 Infinite Set

• Hindi natatapos

• May … (ellipsis)

Example:

• Even numbers = {2,4,6,8,…}

• Multiples of 100 = {100,200,300,…}

🧠 Mnemonic:
Finite = finish
Infinite = forever

6. Set Equality

Two sets are equal if:

• Same elements

• Order doesn’t matter

• Duplicates don’t matter

Examples:

• {9,2,7,-3} = {7,9,-3,2} ✅

• {dog, cat, horse} = {cat, horse, dog, dog} ✅

• {1,2,3} ≠ {1,3,5} ❌

🧠 Rule:
Content > order > repetition

7. Subset (⊆)

A set A is a subset of B if:

• Lahat ng elements ng A ay nasa B

Symbol:

• A ⊆ B

Example:

U = set of all numbers
A = {4,5,6}
➡ A ⊆ U ✅

But:
A = {3, a, 5}
➡ ❌ not a subset (may letters)

🧠 Mnemonic:
Subset = “kasya sa loob”

8. Proper Subset (⊂)

• Subset pero hindi equal

• Kulang ng at least one element

Symbol:

• A ⊂ B

Comparison Table

Example:

• {dog, cat} ⊂ {dog, cat, bird} ✅

• {dog, cat} ⊂ {cat, dog} ❌ (equal lang)

9. Number of Subsets

If a set has n elements:

• Number of subsets = 2ⁿ

• Number of proper subsets = 2ⁿ − 1

Example:

Set = {dog, cat}

• Subsets = 2² = 4

• Proper subsets = 3

🧠 Mnemonic:
Subsets = power of 2
Proper = minus 1

10. Venn Diagram

👉 Visual learner ka? Ito ‘yun.

A Venn diagram:

• Uses circles inside a rectangle (U)

• Shows relationships between sets

🧠 Use this for:

• Union

• Intersection

• Difference

• Word problems (sports, students, etc.)

11. Algebra of Sets (Set Operations)

🔹 Complement (’ or c)

• Elements NOT in the set

• Based on U

Example:

• U = alphabet

• V = vowels
  V’ = consonants

🔹 Intersection (∩)

• Elements common to both sets

• Think: overlap

Example:

• Girls ∩ Adults = adult girls

🔹 Union (∪)

• All elements from both sets

• No duplicates

Example:

• Even ∪ Odd = all numbers

🔹 Set Difference (−)

• A − B = nasa A pero wala sa B

Example:

• A = {1–10}

• B = {2,4,6,8,10}
  A − B = {1,3,5,7,9}

🔹 Symmetric Difference (⊕)

• Elements in either set

• But NOT both

Formula:

• A ⊕ B = (A ∩ B)’

Example:

• A = {3,6,9}

• B = {2,4,6,8,10}
  A ⊕ B = {2,3,4,8,9,10}

🧠 Mnemonic:
Symmetric = “exclusive friendship”

12. Word Problems (Sports Example)

Used ang Venn diagram para sagutin:

• Who plays all sports? → intersection of all

• Who plays none? → complement

• Who plays two sports? → intersection of two circles

📌 Always:

1. Identify U

2. Label sets

3. Fill overlaps first

🔁 FINAL RECAP & MEMORY TRICKS

🧠 Core Ideas

• Set = unordered + well-defined

• Subset = nasa loob

• Proper subset = kulang

• Union = pagsamahin

• Intersection = common

• Difference = tanggal

• Symmetric diff = exclusive

🔑 Exam Survival Tips

• Ignore order & duplicates

• Draw Venn diagrams

• Memorize formulas (2ⁿ, 2ⁿ − 1)

🧠 One-liner Summary