Logic Cheat Sheet

A Education › Mathematics cheat sheet in English by mariusvnh

A simple cheat sheet that describes logical operator and their use

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PRs

Logical Operators

Not	Negation	NOT	¬
And	Conjunction	AND	∧
Or	Disjunction	OR	∨
Exclusive Or	N/A	XOR	⊕
N/A	Implication	IMPLIES	→
Equal	Equivalence	EQUAL	↔

Precedence

NOT

AND, NAND

OR, NOR

IMPLIES

EQUAL, XOR

Commutativity

A op B == B op A
Works for AND, OR, EQUAL, XOR, NAND, and NOR

The only operator that isn't commutative is IMPLIES

Associativity

A op (B op C) == (A op B) op C
Works for AND, OR, EQUAL, and XOR.

Distributivity

A ∨ (B ∧ C)	==	(A ∨ B) ∧ (A ∨ C)
A ∧ (B ∨ C)	==	(A ∧ B) ∨ (A ∧ C)

Implies Truth Table

A	B	Result
False	False	True
True	False	False
False	True	True
True	True	True

Equivalents Formulae

A ↔ B	==	(A → B) ∧ (B → A)
A ⊕ B	==	¬(A → B) ∨ ¬(B → A)
A → B	==	¬A ∨ B
A → B	==	¬(A ∧ ¬B)
A ∨ B	==	¬(¬A ∧ ¬B)
A ∨ B	==	¬A → B
A ∧ B	==	¬(¬A ∨ ¬B)
A ∧ B	==	¬(A → ¬B)

Sementic Tableaux

If you are asked to prove something with sementic tableaux, negate the formulae first.
If you are asked to refute something with the sementic tableaux, do not negate the formulae.

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