MAT101: Algebra and Trigonometry Study Guide

BASIC CONCEPTS OF SET THEORY

• Definition of a Set: A well-defined collection of distinct objects. Sets are denoted by capital letters (e.g., $A, B, C$).
• Methods of Describing Sets:
  • List (Roster) Notation: Elements are listed within braces. Best for finite sets. Example: $A = {2, 4, 6, 8, 10, 12}$.
  • Predicate (Set-Builder) Notation: Stating a property shared by elements. Example: $A = {x : x \text{ is a natural number and less than } 8}$.
  • Recursive (Inductive) Definition: A procedure to generate members. It consists of three parts:
  1. Basis Clause: Establishes the "seeds" or initial objects in the set.
  2. Inductive Clause: Explains how to produce new elements from existing ones.
  3. Extremal Clause: Asserts that only objects obtained via the basis and inductive clauses belong to the set.
  • Example (Natural Numbers $N$):
    • Basis: $1 \in N$
    • Induction: For any element $x \in N$, $x + 1 \in N$
    • Extremal: Only objects from these clauses are in $N$.
• Classification of Sets:
  • Null (Empty) Set: Contains no members. Denoted by $\phi$ or $\emptyset$ or $Reference: { }.
  • Unit Set (Singleton): Contains exactly one member. Example: T = {0}.
  • Finite Set: A set whose members are countable or countably infinite (can be put in one-to-one correspondence with natural numbers).
  • Infinite Set: A set whose members are uncountable and infinitely many.
  • Universal Set (U): The collection of all objects in a particular context.
• Subsets and Supersets:
  • Subset (A \subseteq B$)</strong>: Every member of$A$is also in$B.
  • Proper Subset (A \subset B$)</strong>:$A \subseteq B$and$A \neq B.
  • Improper Subsets: For any set A$, the null set$\phi$and the set$A itself are improper (or trivial) subsets.
  • Power Set (P(S)$)</strong>: The set of all possible subsets of$S$. Formally:$P(S) = { A : A \subseteq S }.
• Cardinality and Identity:
  • Cardinality (n(A)): The number of distinct elements in a finite set.
  • Equivalent Sets (A \approx B$)</strong>: Two sets with the same cardinality:$n(A) = n(B).
  • Equal Sets (A = B): Two sets with the exact same elements. Note: Equal sets are always equivalent, but equivalent sets are not necessarily equal.
  • Complimentary Set (A'$)</strong>: Elements in the universal set$U$that are not in$A$.$A' = {x \in U : x \notin A}.

OPERATIONS ON SETS

• Union (A \cup B$)</strong>: The set of elements in$A$, in$B$, or both.$A \cup B = {x : x \in A \text{ or } x \in B}.
• Intersection (A \cap B$)</strong>: The set of elements common to both.$A \cap B = {x : x \in A \text{ and } x \in B}.
• Difference (A - B$)</strong>: Known as "A minus B" or relative complement. Elements found in$A$but not in$B$.$A - B = {x : x \in A \text{ and } x \notin B}.
• Venn Diagrams: Illustrations used to represent relationships and operations between sets visually.

PROPERTIES OF OPERATIONS ON SETS

• Commutative Properties:
  • Union: A \cup B = B \cup A
  • Intersection: A \cap B = B \cap A
• Associative Properties:
  • Union: A \cup (B \cup C) = (A \cup B) \cup C
  • Intersection: A \cap (B \cap C) = (A \cap B) \cap C
• Distributive Properties:
  • Union over Intersection: A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
  • Intersection over Union: A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
• Idempotent Laws:
  • A \cup A = A
  • A \cap A = A
• Identity Laws:
  • A \cup \phi = A
  • A \cup U = U
  • A \cap \phi = \phi
  • A \cap U = A
• Complement Laws:
  • A \cup A' = U
  • (A')' = A
  • A \cap A' = \phi
  • A - B = A \cap B'
• De Morgan's Laws:
  • For two sets: (A \cup B)' = A' \cap B'$and$(A \cap B)' = A' \cup B'
  • For three sets: (A \cup B \cup C)' = A' \cap B' \cap C'$and$(A \cap B \cap C)' = A' \cup B' \cup C'

CARTESIAN PRODUCT OF SETS

• Ordered Pairs: Elements where order matters: (a, b) \neq (b, a)$unless$a = b.
• Definition: The Cartesian product A \times B$is the set of all ordered pairs$(a, b)$where$a \in A$and$b \in B$. Example: If$A={2,3}, B={4,5}$, then$A \times B = {(2,4), (2,5), (3,4), (3,5)}.
• Cardinality: If set A$has$m$elements and set$B$has$n$elements, then$A \times B$has$m \times n elements.
• Binary Relation: A subset of the Cartesian product. A relation from A$to$B$is a subset of$A \times B$. The total number of relations is$2^{mn}.
• Theorems:
  • A \times \phi = \phi
  • A \times (B \cap C) = (A \times B) \cap (A \times C)
  • A \times (B \cup C) = (A \times B) \cup (A \times C)

THE REAL NUMBER SYSTEM

• Natural Numbers (N$)</strong>: Counting numbers$\dots {1, 2, 3, \dots}. Closed under addition and multiplication.
• Whole Numbers (W$)</strong>: Natural numbers plus zero$\dots {0, 1, 2, 3, \dots}.
  • Zero (0$)</strong>: Additive identity ($a + 0 = a). Acts as a placeholder in the place-value system.
• Integers (Z$)</strong>: Whole numbers and their negatives$\dots {\dots, -2, -1, 0, 1, 2, \dots}. Closed under addition, subtraction, and multiplication, but not division.
• Rational Numbers (Q$)</strong>: Numbers in the form$\frac{a}{b}$where$a, b \in Z$and$b \neq 0.
  • Decimals of rational numbers are either terminating (e.g., 0.5$) or <strong>repeating/recurring</strong> (e.g.,$0.333\dots).
  • Positive Rational: ab > 0$. <strong>Negative Rational</strong>:$ab < 0.
• Irrational Numbers (Q_I$)</strong>: Decimals that do not terminate or recur (e.g.,$\sqrt{2}, \pi). Surds are a subset of irrational numbers.
• Real Numbers (R$)</strong>: The union of rational and irrational numbers ($Q \cup Q_I).
• Properties of Real Numbers:
  • Trichotomy Law: For real numbers a, b$, exactly one is true:$a < b, a = b, a > b.
  • Transitive Law: If a < b$and$b < c$, then$a < c.
  • Additive Identity: 0$. <strong>Multiplicative Identity</strong>:$1.
  • Inverses: Additive inverse (-a$) such that$a + (-a) = 0$; Multiplicative inverse ($\frac{1}{a}$where$a \neq 0$) such that$a \times \frac{1}{a} = 1.
  • BODMAS: Order of operations: Brackets, Of, Division, Multiplication, Addition, Subtraction.

OPERATIONS ON MATRICES

• Definition: A rectangular array of numbers arranged in m$rows and$n$columns ($m \times n).
• Equality: Equal if same size and all corresponding elements are identical (a{ij} = b{ij}).
• Special Matrices:
  • Identity Matrix (I$)</strong>: Square matrix with$1s$on the main diagonal and$0s elsewhere.
  • Zero Matrix (0): All elements are zero.
  • Transpose Matrix (A^T$)</strong>: Obtained by interchanging rows and columns ($A^T{ij} = a{ji}).
  • Diagonal Matrix: Non-zero elements exist only on the main diagonal.
• Addition and Subtraction: Defined only for matrices of the same size. Sum/difference is obtained by adding/subtracting corresponding elements.
• Scalar Multiplication: Multiplying every element of a matrix by a scalar k.
• Multiplication: Product of an m \times n$matrix$A$and an$n \times p$matrix$B$results in an$m \times p$matrix. Note: Matrix multiplication is generally <strong>not commutative</strong> ($AB \neq BA).
• Determinants:
  • For 2 \times 2$Matrix$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$,$\det(A) = ad - bc.
  • For 3 \times 3 Matrix, calculated using minors and cofactors along a row or column.
• Inverse (A^{-1}$)</strong>:$AA^{-1} = I$. Only exists if$\det(A) \neq 0. Formulas:
  • Formula: A^{-1} = \frac{1}{\det(A)} \text{adj}(A), where adj(A) is the transpose of the matrix of cofactors.
• Systems of Equations: Linear equations can be represented as AX = B$. Solution:$X = A^{-1}B.

POLYNOMIAL FUNCTIONS AND LINEAR PROGRAMMING

• Relations: Sets of ordered pairs. Domain is the set of first elements; Range is the set of second elements.
• Functions: A relation where each first coordinate has exactly one second coordinate (survives the Vertical Line Test).
• Linear Function: f(x) = ax + b$.$a$is the gradient (slope),$b is the y-intercept.
• Quadratic Function: f(x) = ax^2 + bx + c$where$a \neq 0. Graph is a parabola.
  • Quadratic Formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • Discriminant (b^2 - 4ac$)</strong>: If$> 0$(2 real roots),$= 0$(equal roots),$< 0 (imaginary roots).
  • Roots Relations: Sum of roots \alpha + \beta = -\frac{b}{a}$; Product of roots$\alpha\beta = \frac{c}{a}.
• Remainder Theorem: If a polynomial f(x)$is divided by$(x - a)$, the remainder is$f(a).
• Factor Theorem: (x - a)$is a factor if and only if$f(a) = 0.
• Absolute Value (|x|$)</strong>: Distance from zero. Equals$x$if$x \geq 0$and$-x$if$x < 0.
• Linear Programming: Optimizing a linear objective function subject to linear constraints (inequalities). Solutions usually occur at the vertices of the feasible region.

INDICES, LOGARITHMS, AND SEQUENCES

• Laws of Indices:
  • a^m \times a^n = a^{m+n}
  • a^m \div a^n = a^{m-n}
  • (a^m)^n = a^{mn}
  • a^{-n} = \frac{1}{a^n}
  • a^{\frac{1}{n}} = \sqrt[n]{a}
• Logarithms: \log_a N = x$implies$a^x = N.
  • Product Rule: \loga(NM) = \loga N + \log_a M
  • Quotient Rule: \loga(\frac{N}{M}) = \loga N - \log_a M
  • Power Rule: \loga(N^p) = p \loga N
• Binomial Theorem (Natural Exponent): (x+y)^n = \sum{k=0}^n C^nk x^{n-k} y^k.
  • Number of terms = n + 1.
• Arithmetic Progression (AP): Constant difference d.
  • n^{th}$term:$u_n = a + (n-1)d.
  • Sum: S_n = \frac{n}{2}[2a + (n-1)d].
• Geometric Progression (GP): Constant ratio r.
  • n^{th}$term:$u_n = ar^{n-1}.
  • Sum: S_n = \frac{a(1-r^n)}{1-r}.
  • Sum to Infinity (|r| < 1$):$S_{\infty} = \frac{a}{1-r}.

TRIGONOMETRY

• Angles: Measured in Degrees or Radians (\pi \text{ rad} = 180^{\circ}).
• Six Functions: \sin, \cos, \tan, \csc, \sec, \cot.
• Identities:
  • \sin^2 x + \cos^2 x = 1
  • 1 + \tan^2 x = \sec^2 x
  • 1 + \cot^2 x = \csc^2 x
• Double Angle Formulas:
  • \sin 2x = 2\sin x \cos x
  • \cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x
  • \tan 2x = \frac{2\tan x}{1 - \tan^2 x}
• Half Angle Formulas:
  • \sin(\frac{u}{2}) = \pm \sqrt{\frac{1 - \cos u}{2}}
  • \cos(\frac{u}{2}) = \pm \sqrt{\frac{1 + \cos u}{2}}
  • \tan(\frac{u}{2}) = \frac{1 - \cos u}{\sin u}$$