Subject
Logic
Statements, truth tables, conditional reasoning, sets, quantifiers, and proof — the foundations of mathematical thinking.
Begin at I if you’re new to Logic.
Statements and Truth Values
A statement is a sentence that's either true or false. Questions, commands, and exclamations are not statements.
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Negation
Negation flips the truth value. ¬T = F and ¬F = T. Two negations cancel.
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Conjunction (AND)
p ∧ q is true only when both p and q are true. Any false part makes the conjunction false.
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Disjunction (OR)
p ∨ q is true when at least one is true. Only false when both are false.
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Conditional Statements
p → q is false only when the hypothesis is true and the conclusion is false.
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Biconditional (If and Only If)
p ↔ q is true when p and q have the same truth value. Two conditionals in one.
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Truth Tables
Truth tables list every combination. Rows = 2^n. Evaluate compounds from the inside out.
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Logical Equivalence
Two statements are equivalent if they share every row of their truth tables. The familiar laws are equivalences.
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De Morgan's Laws
Push a negation through a connective by flipping AND ↔ OR and negating each part.
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Converse, Inverse, Contrapositive
Three transformations of a conditional. Only the contrapositive is always equivalent to the original.
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Valid Argument Forms
MP, MT, HS, DS are valid. Affirming the consequent and denying the antecedent are not.
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Logical Fallacies
Affirming the consequent and denying the antecedent look valid but aren't. Spot them and reject them.
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Deductive vs Inductive Reasoning
Deductive reasoning applies general rules to specific cases — guaranteed if premises are true. Inductive infers patterns from observations — likely, not certain.
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Counterexamples
One specific case is enough to disprove an 'all' or 'every' claim. The trick is finding it.
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Sets and Set Operations
Union, intersection, complement, and difference. Inclusion-exclusion for counting unions.
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Venn Diagrams
Count people or items in specific regions: only-A, both, either, neither.
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Quantifiers
∀ (for all) vs ∃ (there exists). Negation swaps them. The two ways to make claims about whole collections.
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Proof Techniques
Direct, contradiction, contrapositive, induction. The four standard tools for mathematical proof.
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