The Mathematical Symbols Guiding Algorithmic Logic

Hey KISJ coders! 🚀As we continue to explore the vast landscape of computer science, it's important to appreciate the role of mathematics in developing our logical reasoning. Here, we're going to focus on a key aspect that often goes unnoticed but is vital for clear communication in our field: mathematical symbols. These symbols are not just for mathematicians; they're essential tools for all of us in computer science—and don't forget calculus—to express complex ideas succinctly and precisely. Let's take a closer look at the notations that streamline our algorithms and proofs.

Basic Logical Connectives

• ∧ (AND): (p ∧ q) means both p and q must be true.
• ∨ (OR): (p ∨ q) means either p is true, q is true, or both are true.
• ¬ (NOT): ¬p means p is not true.
• → (Implication): (p → q) means if p is true, then q must also be true.
• ↔ (Equivalence): (p ↔ q) means p is true if and only if q is true.

Quantifiers

• ∀ (For all): ∀x ∈ ℝ, x² ≥ 0 means for all real numbers x, the square of x is non-negative.
• ∃ (There exists): ∃x ∈ ℝ, x² = 2 means there exists a real number x such that x squared equals 2.
• ∃! (There exists exactly one): ∃!n ∈ ℕ, n + 5 = 10 means there exists exactly one natural number n such that n plus 5 equals 10.

Set Theory

• ∈ (Element of): 3 ∈ ℕ means 3 is an element of the set of natural numbers.
• ∉ (Not an element of): 2/3 ∉ ℕ means 2/3 is not an element of the set of natural numbers.
• ⊂ (Subset of): ℕ ⊂ ℤ means the set of natural numbers is a subset of the set of integers.
• ⊆ (Subset or equal to): ℤ ⊆ ℤ means the set of integers is a subset of or equal to itself.
• ⊄ (Not a subset of): {2, 3.5} ⊄ ℕ means the set containing 2 and 3.5 is not a subset of the natural numbers.
• ∪ (Union): {1, 2} ∪ {2, 3} = {1, 2, 3} means the union of the sets {1, 2} and {2, 3} is {1, 2, 3}.
• ∩ (Intersection): {1, 2} ∩ {2, 3} = {2} means the intersection of the sets {1, 2} and {2, 3} is {2}.
• ∅ (Empty set): ∅ is the set containing no elements.

Relations

• = (Equality): sin²(θ) + cos²(θ) = 1 is the Pythagorean identity in trigonometry.
• ≠ (Inequality): x² ≠ -1 for any x ∈ ℝ since squares of real numbers are non-negative.
• < (Less than): e^x < e^(x+1) for all x ∈ ℝ.
• > (Greater than): n! > n for any n ∈ ℕ where n > 1.
• ≤ (Less than or equal to): x ≤ x² for all x ∈ [1, ∞).
• ≥ (Greater than or equal to): x² ≥ 0 for all x ∈ ℝ.
• ≈ (Approximately equal to): π ≈ 3.14159.
• ≡ (Identical to or congruent): 7 ≡ 3 (mod 4) means 7 is congruent to 3 modulo 4.

Proof Notation

• ∴ (Therefore): ∀x ∈ ℝ, x² ≥ 0 ∴ x² + 5 ≥ 5.
• ∵ (Because): x² - 9 = 0 ∵ x = ±3.
• ⇒ (Implies): (x² = 4) ⇒ (x = ±2).
• ⇐ (Implied by): (x = ±2) ⇐ (x² = 4).
• ⇔ (If and only if): (n² is even) ⇔ (n is even) for any n ∈ ℤ.
• ∎ (End of proof): After demonstrating a proof, we conclude with ∎.

Miscellaneous

• ... (Ellipsis): Used in sequences, e.g., 1, 1/2, 1/4, 1/8, ... converges to 0.
• : (Such that): {x ∈ ℝ : x > 0} is the set of all x in ℝ such that x is greater than 0.
• ∞ (Infinity): lim (1/n) as n → ∞ = 0.
• ∼ (Is similar to or has the property of): f(x) ∼ g(x) as x → ∞ means f(x) is asymptotically similar to g(x).