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(Propositional Logic Mind Map, Propositional Logic, Definition: Branch ofโฆ
Propositional Logic Mind Map
Propositional Logic
Definition: Branch of logic dealing with propositions and their relationships.
Propositions
Definition: Statements that are either true or false.
Examples: "The sky is blue.", "2 + 2 = 4."
Logical Connectives
AND (โง)
Definition: Conjunction; true if both propositions are true.
Example:
๐
โง
๐
pโงq
OR (โจ)
Definition: Disjunction; true if at least one proposition is true.
Example:
๐
โจ
๐
pโจq
NOT (ยฌ)
Definition: Negation; true if the proposition is false.
Example:
ยฌ
๐
ยฌp
IMPLIES (โ)
Definition: Implication; true if when the first proposition is true, the second one is also true.
Example:
๐
โ
๐
pโq
IF AND ONLY IF (โ)
Definition: Biconditional; true if both propositions are either true or false.
Example:
๐
โ
๐
pโq
Truth Tables
Definition: Tables used to determine the truth value of propositions based on the truth values of their components.
Example: Truth table for
๐
โง
๐
pโงq
Tautologies
Definition: Propositions that are always true, regardless of the truth values of their components.
Example:
๐
โจ
ยฌ
๐
pโจยฌp
Contradictions
Definition: Propositions that are always false.
Example:
๐
โง
ยฌ
๐
pโงยฌp
Contingencies
Definition: Propositions that are neither always true nor always false.
Example:
๐
โ
๐
pโq
Logical Equivalence
Definition: Two propositions that have the same truth value in every possible scenario.
Example:
๐
โ
(
๐
โ
๐
)
โก
(
๐
โง
๐
)
โ
๐
pโ(qโr)โก(pโงq)โr
Rules of Inference
Modus Ponens
Definition: If
๐
โ
๐
pโq and
๐
p are true, then
๐
q is true.
Example:
๐
,
๐
โ
๐
โด
๐
p,pโqโดq
Modus Tollens
Definition: If
๐
โ
๐
pโq and
ยฌ
๐
ยฌq are true, then
ยฌ
๐
ยฌp is true.
Example:
๐
โ
๐
,
ยฌ
๐
โด
ยฌ
๐
pโq,ยฌqโดยฌp
Disjunctive Syllogism
Definition: If
๐
โจ
๐
pโจq and
ยฌ
๐
ยฌp are true, then
๐
q is true.
Example:
๐
โจ
๐
,
ยฌ
๐
โด
๐
pโจq,ยฌpโดq
Hypothetical Syllogism
Definition: If
๐
โ
๐
pโq and
๐
โ
๐
qโr are true, then
๐
โ
๐
pโr is true.
Example:
๐
โ
๐
,
๐
โ
๐
โด
๐
โ
๐
pโq,qโrโดpโr
Applications
Computer Science
Logic Gates
Algorithms
Mathematics
Proofs
Theorems
Philosophy
Logical Reasoning
Argument Analysis