➗ Algebra

What Is Algebra (Really)?

Algebra studies structures defined by operations and rules, not just numbers.

• Objects: numbers, vectors, polynomials, matrices
• Operations: +, ·, composition, etc.
• Rules: associativity, commutativity, distributivity

> Algebra asks: What stays true if we change the objects but keep the rules?

We move from solving single equations to understanding whole systems and structures.

Notation Refresher

• Variables: usually x, y, z
• Parameters: a, b, c; fixed within a problem
• Domains: ℝ, ℚ, ℤ, ℂ
• Functions: f: A → B, f(x)
• Sets: {x ∈ ℝ | condition}

Careful notation prevents logical errors in long, multi-step algebraic arguments.

Expressions vs. Equations vs. Identities

• Expression: 3x² − 2x + 1
• Equation: 3x² − 2x + 1 = 0
• Identity: (x + 1)² ≡ x² + 2x + 1

Identities hold for all allowed values; equations usually hold only for specific solutions. This distinction matters in proof and simplification.

Quantifiers: ∀ and ∃

Algebraic statements often use:

• Universal: ∀x ∈ ℝ, x² ≥ 0
• Existential: ∃x ∈ ℝ such that x² = 2

Order matters:

• ∀x ∃y (y > x) is true in ℝ
• ∃y ∀x (y > x) is false in ℝ

Reading quantifiers carefully is essential preparation for proofs.