Fundamentals of Propositional Logic and Combinatorics

Propositional Logic

• Propositional logic: The study of arguments whose validity depends on how the statements are connected, regardless of their meaning.
• Argument: A sequence of propositions whose purpose is the implication of another proposition.
• Premise: A succession of propositions that serve as evidence.
• Conclusion: The inferred proposition.
• Proposition: A declarative sentence whose meaning can be True or False.
• Variable: An unspecified element of a reference set.
• Propositional form: An expression containing a variable that becomes a proposition when the variable is substituted by an element of a reference set.
• Simple statement: Refers to a single event.
• Tautology: A compound proposition that is True under all possible truth conditions.
• Contradiction: A compound proposition that is False under all possible truth conditions of the propositions it is composed of.
• Contingency: A proposition that is neither a contradiction nor a tautology.
• Logical connective: Logical symbols used to construct compound statements from simple statements.
• Valid argument: An argument is valid if and only if the conjunction of the premises implies the conclusion.
• Solution set: All elements of the reference set that make the proposition True.
• Negation: Given the proposition "p", the negation of "p" is defined as the proposition that is False when "p" is True.
• Propositional calculus: The fundamental operations used to construct complex propositions from simple propositions.

Combinatorics and Algebra

• Newton series (Newton's Binomial): The expression (x + a)ⁿ whose expansion is a homogeneous polynomial of degree n.
• General term of (x + a)ⁿ: T_k+1 = C^k_n a^k x^n-k
• Permutations: All the groups that can be formed with "n" elements of a collection. Two permutations are different if the order of at least two of their elements is changed.
• Arrangements: All groups of k elements that can be formed from a collection of n elements. Two arrangements are different if the order of at least two of their elements is changed.
• Combination: Similar to arrangements, but two combinations are different if they contain at least one different element.
• Complementary arrangement: Two arrangements are complementary when the sum of their classes is equal to the number of elements in the initial collection.
• Complementary combination: Two combinations are complementary when the sum of their classes is equal to the number of elements in the initial collection.
• Properties of combinations:
  1. C^k_n = C^n-k_n
  2. Cⁿ_n = 1
  3. C⁰_n = 1
• Factorial: The factorial of a number "n" (denoted as n!) is defined as the product of all natural numbers from 1 to n.
• Formula linking combinations and arrangements: C^k_n = A^k_n / P_k
• Matrix: A rectangular distribution of elements arranged in rows and columns.