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Cheatography

A simple cheat sheet that describes logical operator and their use

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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