📚 What is a Set?
A set is a well-defined collection of distinct objects. "Well-defined" means there must be no ambiguity about whether an object belongs to the set or not. Sets are usually denoted by capital letters (A, B, C) and elements by lowercase letters (a, b, c).
Notation: If an element x belongs to set A, we write x ∈ A (read "x belongs to A"). If not, x ∉ A.
Representation Methods:
- Roster/Tabular: List all elements in curly braces. A = {1, 2, 3, 4, 5}
- Set-Builder: Describe the property. A = {x : x is a natural number, x ≤ 5} — read as "the set of all x such that x is a natural number less than or equal to 5"
Important Types of Sets:
| Type | Definition | Example |
|---|---|---|
| Empty/Null Set (∅) | Contains NO elements | {x : x is a natural number less than 1} = ∅ or {} |
| Singleton Set | Contains exactly ONE element | {5} |
| Finite Set | Has a countable number of elements | {2, 4, 6, 8} |
| Infinite Set | Has uncountable elements | {1, 2, 3, 4, ...} (natural numbers) |
| Universal Set (U) | Contains ALL elements under consideration | If we discuss integers, U = Z |
| Equal Sets | Same elements: A = B | {1,2,3} = {3,1,2} (order doesn't matter) |
| Subset (A ⊆ B) | Every element of A is also in B | {1,2} ⊆ {1,2,3} |
| Proper Subset (A ⊂ B) | A is subset of B but A ≠ B | {1,2} ⊂ {1,2,3} |
| Power Set P(A) | Set of ALL subsets of A. If |A|=n, then |P(A)|=2ⁿ | P({1,2}) = {∅, {1}, {2}, {1,2}} |
Number Sets You Must Know:
- ℕ (Natural Numbers): {1, 2, 3, 4, ...} — counting numbers starting from 1
- ℤ (Integers): {..., -3, -2, -1, 0, 1, 2, 3, ...} — includes negatives and zero
- ℚ (Rational Numbers): Numbers that can be written as p/q where q ≠ 0. Example: 1/2, 3, -7/4, 0.5
- ℝ (Real Numbers): All rational + irrational numbers. Includes everything on the number line.
📚 Set Operations — in Full Detail
1. Union (A ∪ B): ALL elements that are in A OR B OR both.
A ∪ B = {1, 2, 3, 4, 5} — note: 3 appears only once (no duplicates in sets)
2. Intersection (A ∩ B): ONLY elements that are in BOTH A AND B.
A ∩ B = {3} — only 3 is in both sets
3. Difference (A − B or A \ B): Elements in A that are NOT in B.
A − B = {1, 2} — remove from A anything that appears in B
B − A = {5} — note: A−B ≠ B−A
4. Complement (A' or Aᶜ): All elements in the Universal Set U that are NOT in A.
A' = {1,3,5,7,9,10}
5. Symmetric Difference (A Δ B): Elements in A or B but NOT in both. Formula: A Δ B = (A − B) ∪ (B − A)
A Δ B = {1, 2, 4, 5} — 3 is removed because it's in BOTH
Counting Formula (for two sets):
Solution: |M ∪ S| = 30 + 20 − 10 = 40 students
For three sets:
De Morgan's Laws (very important!):
Memory trick: "Break the bar (complement), change the sign (∪ becomes ∩ and vice versa)"
Properties of Set Operations:
| Property | Union | Intersection |
|---|---|---|
| Commutative | A∪B = B∪A | A∩B = B∩A |
| Associative | (A∪B)∪C = A∪(B∪C) | (A∩B)∩C = A∩(B∩C) |
| Distributive | A∪(B∩C) = (A∪B)∩(A∪C) | A∩(B∪C) = (A∩B)∪(A∩C) |
| Identity | A∪∅ = A | A∩U = A |
| Complement | A∪A' = U | A∩A' = ∅ |
| Idempotent | A∪A = A | A∩A = A |
📚 Relations & Functions
A relation R from set A to set B is a subset of A × B (the Cartesian product). Informally, it defines how elements of A are related to elements of B.
A × B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)} — all possible ordered pairs
R = {(1,a),(2,b),(3,a)} — this is a relation (subset of A×B)
Types of Relations (on a set A):
| Type | Definition | Example on {1,2,3} |
|---|---|---|
| Reflexive | Every element is related to itself: (a,a) ∈ R for all a | {(1,1),(2,2),(3,3),...} — must contain all diagonal pairs |
| Symmetric | If (a,b) ∈ R then (b,a) ∈ R | If (1,2) is in R, then (2,1) must also be |
| Transitive | If (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R | If (1,2) and (2,3) in R, then (1,3) must be too |
| Equivalence | Reflexive + Symmetric + Transitive | "=" (equality) is an equivalence relation |
| Anti-symmetric | If (a,b) and (b,a) ∈ R then a = b | "≤" is anti-symmetric |
Functions: A function f: A → B is a special relation where every element of A maps to exactly ONE element of B. In other words, no element of A can have two different outputs.
| Type | Definition | Visual Test |
|---|---|---|
| One-to-One (Injective) | Different inputs → Different outputs. If f(a)=f(b) then a=b. | No two arrows from A point to the same element in B |
| Onto (Surjective) | Every element of B has at least one element mapping to it from A. | Every element in B has at least one arrow pointing to it |
| Bijective | Both one-to-one AND onto. Perfect pairing. | Exactly one arrow to each element of B, none left out |
• One-to-one? Yes. If 2a+1 = 2b+1, then a=b.
• Onto? Yes. For any y, x = (y−1)/2 maps to it.
• Bijective? Yes! It is both injective and surjective.
📚 What is a Matrix?
A matrix is a rectangular arrangement of numbers in rows and columns, enclosed in brackets. Matrices are used to solve systems of equations, perform transformations in graphics, and represent data in computer science.
A matrix with m rows and n columns is called an m × n matrix. The element in row i and column j is written as aᵢⱼ.
Types of Matrices:
| Type | Definition | Example (if applicable) |
|---|---|---|
| Row Matrix | Only 1 row (1×n) | [1 2 3] |
| Column Matrix | Only 1 column (m×1) | 3 rows, 1 column |
| Square Matrix | Rows = Columns (n×n) | 2×2, 3×3, etc. |
| Zero/Null Matrix | All elements are 0 | Every entry is 0 |
| Diagonal Matrix | Non-zero only on main diagonal | aᵢⱼ = 0 when i ≠ j |
| Identity Matrix (I) | Diagonal matrix with all 1s on diagonal | Like "1" for multiplication: AI = A |
| Symmetric | A = Aᵀ (equals its transpose) | aᵢⱼ = aⱼᵢ |
| Skew-Symmetric | Aᵀ = −A, diagonal elements = 0 | aᵢⱼ = −aⱼᵢ |
| Upper Triangular | All elements below diagonal = 0 | |
| Lower Triangular | All elements above diagonal = 0 |
📚 Matrix Operations
Addition/Subtraction: Add/subtract corresponding elements. Matrices must be the same size.
Scalar Multiplication: Multiply every element by the scalar.
Matrix Multiplication (A × B): The most important operation. A must be m×n and B must be n×p (columns of A = rows of B). Result is m×p.
Each element of the result = dot product of a row of A with a column of B.
A = [1 2; 3 4] (2×2) × B = [5 6; 7 8] (2×2)
Result[1][1] = (1×5) + (2×7) = 5 + 14 = 19
Result[1][2] = (1×6) + (2×8) = 6 + 16 = 22
Result[2][1] = (3×5) + (4×7) = 15 + 28 = 43
Result[2][2] = (3×6) + (4×8) = 18 + 32 = 50
A × B = [19 22; 43 50]
Transpose (Aᵀ): Swap rows and columns. Row i becomes column i.
Properties of Transpose:
- (Aᵀ)ᵀ = A
- (A + B)ᵀ = Aᵀ + Bᵀ
- (kA)ᵀ = kAᵀ
- (AB)ᵀ = BᵀAᵀ (order reverses!)
📚 Determinants
A determinant is a single number calculated from a square matrix. It tells us:
- Whether the matrix has an inverse (det ≠ 0 means inverse exists)
- Area/volume scaling in geometry
- Whether a system of equations has a unique solution
2×2 Determinant:
3×3 Determinant (Expansion along first row):
= 1(5×9 − 6×8) − 2(4×9 − 6×7) + 3(4×8 − 5×7)
= 1(45−48) − 2(36−42) + 3(32−35)
= 1(−3) − 2(−6) + 3(−3)
= −3 + 12 − 9 = 0
Inverse of a Matrix: A⁻¹ exists only if det(A) ≠ 0.
For a 2×2 matrix:
A⁻¹ = (1/10) × [4 −2; −1 3] = [0.4 −0.2; −0.1 0.3]
Verify: A × A⁻¹ should equal the Identity matrix I.
Cramer's Rule — Solving systems of equations using determinants:
D = |2 3; 1 4| = 8 − 3 = 5
Dₓ = |8 3; 7 4| = 32 − 21 = 11 → x = 11/5 = 2.2
Dᵧ = |2 8; 1 7| = 14 − 8 = 6 → y = 6/5 = 1.2
📚 Limits — The Foundation of Calculus
A limit asks: "What value does f(x) approach as x gets closer and closer to some number?" We write: lim(x→a) f(x) = L.
The function doesn't need to actually equal L at x=a. It just needs to approach L.
If we plug x=2 directly: (4−4)/(2−2) = 0/0 (undefined!)
But factoring: (x²−4)/(x−2) = (x+2)(x−2)/(x−2) = x+2
So lim = 2+2 = 4
Key Limit Rules:
- lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
- lim [f(x) × g(x)] = lim f(x) × lim g(x)
- lim [f(x)/g(x)] = lim f(x) / lim g(x) (if denominator ≠ 0)
- lim [k × f(x)] = k × lim f(x)
Standard Limits (memorize!):
lim(x→0) (1 − cos x)/x = 0
lim(x→0) (eˣ − 1)/x = 1
lim(x→0) log(1+x)/x = 1
lim(x→∞) (1 + 1/x)ˣ = e ≈ 2.71828
📚 Derivatives — Rate of Change
The derivative of a function f(x) gives the rate at which f(x) changes as x changes. Geometrically, it's the slope of the tangent line at any point.
All Derivative Formulas You Need:
| Function f(x) | Derivative f'(x) | Example |
|---|---|---|
| xⁿ | nxⁿ⁻¹ | d/dx(x³) = 3x² |
| constant (c) | 0 | d/dx(7) = 0 |
| eˣ | eˣ | Unique! It's its own derivative |
| aˣ | aˣ · ln(a) | d/dx(2ˣ) = 2ˣ ln(2) |
| ln(x) | 1/x | d/dx(ln x) = 1/x |
| sin(x) | cos(x) | |
| cos(x) | −sin(x) | Note the negative! |
| tan(x) | sec²(x) |
Derivative Rules:
| Rule | Formula | Example |
|---|---|---|
| Sum/Difference | (f±g)' = f' ± g' | d/dx(x³ + 5x) = 3x² + 5 |
| Constant Multiple | (cf)' = c·f' | d/dx(7x²) = 14x |
| Product Rule | (fg)' = f'g + fg' | d/dx(x²·sin x) = 2x·sin x + x²·cos x |
| Quotient Rule | (f/g)' = (f'g − fg')/g² | d/dx(sin x/x) = (cos x · x − sin x)/x² |
| Chain Rule | d/dx[f(g(x))] = f'(g(x)) · g'(x) | d/dx(sin(3x)) = cos(3x) · 3 = 3cos(3x) |
Find d/dx [x³ · eˣ]
Let f = x³, g = eˣ
f' = 3x², g' = eˣ
(fg)' = f'g + fg' = 3x²·eˣ + x³·eˣ = eˣ(3x² + x³) = x²eˣ(3 + x)
Applications of Derivatives:
- Finding maxima/minima: Set f'(x) = 0, solve for x. These are critical points. Use second derivative test: f''(x) > 0 → minimum, f''(x) < 0 → maximum.
- Rate of change: Velocity = derivative of position, Acceleration = derivative of velocity.
- Increasing/Decreasing: f'(x) > 0 means function is increasing, f'(x) < 0 means decreasing.
📚 Integration — The Reverse of Differentiation
If differentiation finds the rate of change, integration finds the original function from its rate of change. It also calculates areas under curves.
The C is the "constant of integration" because many functions can have the same derivative (e.g., x²+5 and x²+100 both have derivative 2x).
All Integration Formulas:
| Function | Integral |
|---|---|
| ∫ xⁿ dx | xⁿ⁺¹/(n+1) + C (n ≠ −1) |
| ∫ 1/x dx | ln|x| + C |
| ∫ eˣ dx | eˣ + C |
| ∫ aˣ dx | aˣ/ln(a) + C |
| ∫ sin x dx | −cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec²x dx | tan x + C |
| ∫ k dx | kx + C |
∫ (3x² + 2x + 5) dx = 3·x³/3 + 2·x²/2 + 5x + C = x³ + x² + 5x + C
∫ (4x³ − 2/x) dx = 4·x⁴/4 − 2·ln|x| + C = x⁴ − 2ln|x| + C
Definite Integrals: Have upper and lower limits — give a number (the area).
📚 Probability — Complete Guide
Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain).
Key Terms:
- Experiment: An activity with uncertain outcome (rolling a die, tossing a coin)
- Sample Space (S): Set of ALL possible outcomes. Die: S = {1,2,3,4,5,6}
- Event: A subset of the sample space. "Getting even" = {2,4,6}
- Complement P(E'): P(event NOT happening) = 1 − P(E)
S = {1,2,3,4,5,6} → |S| = 6
Favorable = {5,6} → |E| = 2
P(E) = 2/6 = 1/3
Addition Rule (OR — at least one event):
If A and B are mutually exclusive (can't happen together): P(A ∪ B) = P(A) + P(B)
Multiplication Rule (AND — both events):
If A and B are independent (one doesn't affect the other): P(A ∩ B) = P(A) × P(B)
Conditional Probability:
Bayes' Theorem:
Permutations & Combinations:
| Permutation (Order Matters) | Combination (Order Doesn't) | |
|---|---|---|
| Formula | P(n,r) = n!/(n−r)! | C(n,r) = n!/[r!(n−r)!] |
| Example | Arrange 3 from 5 people in a line | Choose 3 from 5 for a committee |
| Calculation | P(5,3) = 5!/2! = 60 | C(5,3) = 5!/(3!·2!) = 10 |
10 × 10 × 10 × 10 = 10,000
Without repetition: 10 × 9 × 8 × 7 = 5,040 (each digit used once)
📚 Statistics — Measures of Central Tendency & Dispersion
Mean (Average):
Median: The middle value when data is sorted. If even count, average the two middle values.
Sorted: 3, 5, 7, 9, 11, 15 → Median = (9+11)/2 = 10
Mode: The value that appears MOST frequently.
Variance & Standard Deviation:
Standard Deviation (σ) = √Variance
SD tells you how spread out the data is. Small SD = data clustered near mean. Large SD = data widely spread.
Mean = (2+4+6+8+10)/5 = 30/5 = 6
Deviations: (2−6)²=16, (4−6)²=4, (6−6)²=0, (8−6)²=4, (10−6)²=16
Variance = (16+4+0+4+16)/5 = 40/5 = 8
SD = √8 = 2.83