The Foundations: Logic and Proofs

Explain what a proposition is in propositional logic

A proposition is a declarative sentence that is either true or false within propositional logic, such as "The Moon is made of green cheese" or "Trenton is the capital of New Jersey" ​

Describe the difference between propositions and sentences that are not propositions

Propositions are statements with a definite truth value (true or false), whereas sentences like commands, questions, or expressions containing variables (e.g., "Sit down!" or "x > 1") are not propositions because their truth value is undefined ​

Explain how compound propositions are constructed

Compound propositions are formed by connecting simpler propositions using logical connectives such as negation, conjunction, disjunction, implication, and biconditional ​

Describe the function and truth table of negation

Negation takes a proposition p and transforms it into ¬p, which is true when p is false and false when p is true; this operation reverses the truth value of the original proposition ​

Compare conjunction and disjunction in propositional logic

Conjunction combines two propositions p and q using "and," resulting in true only if both are true; disjunction uses "or" and is true if at least one proposition is true ​

Analyze the difference between inclusive or and exclusive or

Inclusive or is true when either or both operands are true, while exclusive or (xor) is true only if exactly one of the operands is true but not both ​

Explain the meaning and components of an implication in logic

An implication "if p, then q" connects two propositions, where p is the antecedent and q is the consequent; it is false only when p is true and q is false, and otherwise true ​

Describe different ways of expressing implication in English

Implications can be phrased as "if p, then q," "p only if q," "q unless p," "q when p," "whenever p, q" and "p is sufficient for q," showing the versatility of conditional statements ​

Compare converse, contrapositive, and inverse of an implication

The converse reverses the implication to "if q then p", the inverse negates both sides "if not p then not q", and the contrapositive reverses and negates "if not q then not p"; only the contrapositive is logically equivalent to the original implication ​

Explain the biconditional operator and its truth table

The biconditional operator (p if and only if q) is true when p and q have the same truth value, and false otherwise, representing logical equivalence in both directions ​

Describe how truth tables are constructed for compound propositions

Truth tables list all possible combinations of truth values for atomic propositions, with columns for each compound expression, allowing evaluation of the overall truth of logical statements ​

Explain how to determine the number of rows in a truth table with n variables

A truth table for n propositional variables has 2n rows, representing every possible combination of true and false values for those variables

Analyze the concept of logical equivalence between propositions

Two propositions are logically equivalent if their truth values match for all possible inputs; this can be shown by comparing columns in their respective truth tables or by demonstrating equivalence through logical laws ​

Justify why neither the converse nor inverse of an implication are generally logically equivalent to the original implication

Truth tables reveal that the converse and inverse of an implication do not consistently match the original's truth values, and thus are not logically equivalent except in special cases ​

Describe common logical operator precedence in propositional logic

Logical operators have specific precedence rules, typically with negation highest, followed by conjunction, disjunction, implication, and biconditional, affecting evaluation order unless parentheses dictate otherwise ​

Explain methods for translating English sentences into propositional logic

To translate, identify atomic propositions, assign variables, and use logical connectives to construct a formal representation of the sentence's meaning ​

Analyze the concept of consistent system specifications using propositional logic

A set of system specifications written as propositions is consistent if there is an assignment of truth values that makes all specifications true simultaneously ​

Describe how logic puzzles use propositional reasoning

Logic puzzles often use statements with specific truth values, such as "knights always tell the truth and knaves always lie," requiring analysis and assignment of truth values to solve them ​

Explain how logic circuits correspond to propositional logic

Electronic circuits like NOT, OR, and AND gates mirror the actions of negation, disjunction, and conjunction in logic, processing binary inputs to produce outputs based on logical rules ​

Contrast tautology, contradiction, and contingency in propositional logic

A tautology is always true regardless of input, a contradiction is always false, and a contingency can be true or false depending on the input values ​

Explain what it means for compound propositions to be logically equivalent

Logical equivalence between compound propositions means that their truth tables yield identical results for all possible combinations of their variables ​

Describe De Morgan’s Laws and their importance in propositional logic

De Morgan’s Laws provide equivalences for the negation of conjunction and disjunction, allowing expressions like ¬(p ∨ q) to be rewritten as (¬p) ∧ (¬q), which simplifies logic manipulation ​

Explain the process of constructing new logical equivalences

New equivalences are proven by transforming one logical expression to another using a sequence of logical identities or truth table analysis until both expressions are shown to be equivalent ​

Describe the Double Negation Law in logic

The Double Negation Law states that negating a proposition twice returns the original value, symbolically ¬(¬p) = p