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Mathematical Logic

Mathematical Logic Formulas

Introduction to Mathematical Logic Section


Mathematical logic studies the principles of reasoning and provides a rigorous framework to analyze the structure of statements and arguments. It begins with the basics of propositional logic, where statements are combined using logical operators like and, or, and not, to form compound statements and evaluate their truth.

A deeper layer is predicate logic, which extends propositional logic by introducing quantifiers like for all and there exists, allowing reasoning about objects and their properties. Central to this is the concept of logical validity, which examines whether conclusions follow from premises regardless of specific interpretations.

Mathematical logic also explores formal systems, which consist of axioms, rules of inference, and symbols for constructing proofs. Key topics include set theory, the foundation of mathematics, and model theory, which studies the relationship between formal languages and their interpretations.

Another significant area is computability theory, which asks fundamental questions about what problems can be solved by algorithms, and proof theory, which investigates the nature and structure of mathematical proofs.

Applications of mathematical logic are vast, influencing fields like computer science, where it underpins algorithms and programming languages, and philosophy, where it sharpens reasoning. It develops skills in abstraction, critical thinking, and formal reasoning, making it a cornerstone of mathematical rigor.

Mathematical Logic Formulas

Explore Mathematical Logic formulas with explanations and examples

Double Negation Law

¬(¬P)=P\neg(\neg P) = P

De Morgan's Law (Conjunction)

¬(P∧Q)=¬P∨¬Q\neg(P \land Q) = \neg P \lor \neg Q

De Morgan's Law (Disjunction)

¬(P∨Q)=¬P∧¬Q\neg(P \lor Q) = \neg P \land \neg Q

Implication as Disjunction

P→Q=¬P∨QP \rightarrow Q = \neg P \lor Q

Contrapositive

P→Q=¬Q→¬PP \rightarrow Q = \neg Q \rightarrow \neg P

Biconditional Definition

P↔Q=(P→Q)∧(Q→P)P \leftrightarrow Q = (P \rightarrow Q) \land (Q \rightarrow P)

Distributive Law (Conjunction over Disjunction)

P∧(Q∨R)=(P∧Q)∨(P∧R)P \land (Q \lor R) = (P \land Q) \lor (P \land R)

Distributive Law (Disjunction over Conjunction)

P∨(Q∧R)=(P∨Q)∧(P∨R)P \lor (Q \land R) = (P \lor Q) \land (P \lor R)

Identity Law (Conjunction)

P∧True=PP \land \text{True} = P

Identity Law (Disjunction)

P∨False=PP \lor \text{False} = P

Domination Law (Conjunction)

P∧False=FalseP \land \text{False} = \text{False}

Domination Law (Disjunction)

P∨True=TrueP \lor \text{True} = \text{True}

Idempotent Law (Conjunction)

P∧P=PP \land P = P

Idempotent Law (Disjunction)

P∨P=PP \lor P = P

Negation of Quantifiers (Universal)

¬(∀x P(x))=∃x ¬P(x)\neg (\forall x\, P(x)) = \exists x\, \neg P(x)

Negation of Quantifiers (Existential)

¬(∃x P(x))=∀x ¬P(x)\neg (\exists x\, P(x)) = \forall x\, \neg P(x)

Universal Instantiation

∀x P(x)→P(c)\forall x\, P(x) \rightarrow P(c)

Existential Generalization

P(c)→∃x P(x)P(c) \rightarrow \exists x\, P(x)

Modus Ponens

From(P→Q)andP,inferQFrom (P \rightarrow Q) and P, infer Q

Modus Tollens

From(P→Q)and¬Q,infer¬PFrom (P \rightarrow Q) and \neg Q, infer \neg P

Hypothetical Syllogism

From(P→Q)and(Q→R),infer(P→R)From (P \rightarrow Q) and (Q \rightarrow R), infer (P \rightarrow R)

Disjunctive Syllogism

From(P∨Q)and¬P,inferQFrom (P \lor Q) and \neg P, infer Q

Law of Excluded Middle

P∨¬P=TrueP \lor \neg P = \text{True}

Law of Non-Contradiction

P∧¬P=FalseP \land \neg P = \text{False}

Double Negation Law

¬(¬P)=P\neg(\neg P) = P

De Morgan's Law (Conjunction)

¬(P∧Q)=¬P∨¬Q\neg(P \land Q) = \neg P \lor \neg Q

De Morgan's Law (Disjunction)

¬(P∨Q)=¬P∧¬Q\neg(P \lor Q) = \neg P \land \neg Q

Implication as Disjunction

P→Q=¬P∨QP \rightarrow Q = \neg P \lor Q

Contrapositive

P→Q=¬Q→¬PP \rightarrow Q = \neg Q \rightarrow \neg P

Biconditional Definition

P↔Q=(P→Q)∧(Q→P)P \leftrightarrow Q = (P \rightarrow Q) \land (Q \rightarrow P)

Distributive Law (Conjunction over Disjunction)

P∧(Q∨R)=(P∧Q)∨(P∧R)P \land (Q \lor R) = (P \land Q) \lor (P \land R)

Distributive Law (Disjunction over Conjunction)

P∨(Q∧R)=(P∨Q)∧(P∨R)P \lor (Q \land R) = (P \lor Q) \land (P \lor R)

Identity Law (Conjunction)

P∧True=PP \land \text{True} = P

Identity Law (Disjunction)

P∨False=PP \lor \text{False} = P

Domination Law (Conjunction)

P∧False=FalseP \land \text{False} = \text{False}

Domination Law (Disjunction)

P∨True=TrueP \lor \text{True} = \text{True}

Idempotent Law (Conjunction)

P∧P=PP \land P = P

Idempotent Law (Disjunction)

P∨P=PP \lor P = P

Negation of Quantifiers (Universal)

¬(∀x P(x))=∃x ¬P(x)\neg (\forall x\, P(x)) = \exists x\, \neg P(x)

Negation of Quantifiers (Existential)

¬(∃x P(x))=∀x ¬P(x)\neg (\exists x\, P(x)) = \forall x\, \neg P(x)

Universal Instantiation

∀x P(x)→P(c)\forall x\, P(x) \rightarrow P(c)

Existential Generalization

P(c)→∃x P(x)P(c) \rightarrow \exists x\, P(x)

Modus Ponens

From(P→Q)andP,inferQFrom (P \rightarrow Q) and P, infer Q

Modus Tollens

From(P→Q)and¬Q,infer¬PFrom (P \rightarrow Q) and \neg Q, infer \neg P

Hypothetical Syllogism

From(P→Q)and(Q→R),infer(P→R)From (P \rightarrow Q) and (Q \rightarrow R), infer (P \rightarrow R)

Disjunctive Syllogism

From(P∨Q)and¬P,inferQFrom (P \lor Q) and \neg P, infer Q

Law of Excluded Middle

P∨¬P=TrueP \lor \neg P = \text{True}

Law of Non-Contradiction

P∧¬P=FalseP \land \neg P = \text{False}