BCA TU, Math I – Unit 1: Set Theory and Real & Complex Numbers Exam Notes

1. Set Theory

Concept, Notation, and Specification of Sets

• Definition: A set is a collection of well-defined and distinct objects.
• Notation: { } (Curly brackets are used to define sets). Example: A = {1, 2, 3, 4}
• Ways to define a set:
  1. Roster/Tabular form: {1, 2, 3, 4}
  2. Set-builder form: {x | x is a natural number, x ≤ 4}

Types of Sets

• Finite Set: {1, 2, 3} (Countable elements)
• Infinite Set: {1, 2, 3, ...} (Uncountable)
• Null Set: { } or ∅ (No elements)
• Universal Set (U): Contains all possible elements under discussion.
• Subset (⊆): If all elements of A exist in B, then A is a subset of B.
• Power Set (P(A)): The set of all subsets of a set.
• Equal Sets: If A and B contain the same elements.

2. Operations on Sets and Venn Diagrams

Basic Set Operations

1. Union (A ∪ B): Elements in A, B, or both.
   • Formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
   • Example: {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
2. Intersection (A ∩ B): Common elements of A and B.
   • Example: {1, 2, 3} ∩ {3, 4, 5} = {3}
3. Difference (A – B): Elements in A but not in B.
   • Example: {1, 2, 3} - {3, 4, 5} = {1, 2}
4. Complement (A’): Elements in Universal Set (U) but not in A.
   • Example: If U = {1, 2, 3, 4, 5} and A = {1, 2}, then A’ = {3, 4, 5}

Laws of Algebra of Sets (Without Proof)

1. Commutative Law:
   • A ∪ B = B ∪ A, A ∩ B = B ∩ A
2. Associative Law:
   • (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C)
3. Distributive Law:
   • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
   • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Cardinal Number of a Set

• Definition: The number of elements in a set, denoted as n(A).
• Formula for three sets: mathematicaCopyEditn(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)
• Example:
  • If n(A) = 5, n(B) = 6, n(A ∩ B) = 2, then n(A ∪ B) = 5 + 6 - 2 = 9.

3. Real Number System

• Real Numbers (R): Includes rational and irrational numbers.
• Rational Numbers (Q): Numbers that can be expressed as a fraction (e.g., 1/2, 3.14).
• Irrational Numbers (Q’): Cannot be expressed as fractions (e.g., √2, π).

Intervals on the Number Line

• Open Interval (a, b): a < x < b (Excludes endpoints).
• Closed Interval [a, b]: a ≤ x ≤ b (Includes endpoints).
• Half-open Interval [a, b): a ≤ x < b.

Absolute Value of a Real Number

• Definition: Distance from 0 on the number line.
• Notation: |x|.
• Example: | -5 | = 5, |3| = 3.

4. Complex Numbers

Definition and Representation

• Complex Number:z = a + ib, where:
  • a = Real part
  • b = Imaginary part (i = √-1)
• Example: 5 + 2i, -3 - i.

Geometrical Representation (Argand Plane)

• X-axis → Real part (Re(z))
• Y-axis → Imaginary part (Im(z))
• Example:z = 3 + 4i is plotted at (3,4).

5. Algebraic Properties of Complex Numbers

Addition

• Formula: (a + ib) + (c + id) = (a + c) + i(b + d)
• Example: (3 + 2i) + (1 + 4i) = 4 + 6i.

Multiplication

• Formula: (a + ib) × (c + id) = (ac - bd) + i(ad + bc).
• Example: (2 + 3i) × (4 - i) = (8 - 3) + i(2 × (-1) + 3 × 4) = 5 + 10i.

Inverse (Reciprocal of Complex Number)

• Formula: 1 / (a + ib) = (a - ib) / (a² + b²).
• Example: 1 / (2 + 3i) = (2 - 3i) / (4 + 9) = (2 - 3i) / 13.

Absolute Value (Modulus) of a Complex Number

• Formula: |z| = √(a² + b²).
• Example: |3 + 4i| = √(3² + 4²) = √(9 + 16) = 5.

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