Boolean algebra is an algebra where there are only two values, typically represented by 1 or 0.^{[1][2]} If the elements are truth values, 1 represents 'true', and 0 represents 'false'.
In boolean algebra, there are four commonly used operators: ∧ (and), ∨ (or), ⊖ (exclusive or), and ¬ (negation, not, or complement). Table 1 summarizes the boolean operators.
Operator  Name  Truth Table  Venn Diagram  Description  

∧  and 

∧ returns true (1) if both operands are true (1), otherwise it returns false (0). In most computer languages, and is represented by '&' or '&&'. The operator '^' represents exponentiation in most computer languages.  
∨  or 

∨ returns true (1) if one or both operands are true (1), otherwise it returns false (0). In most computer languages, or is represented by '' or ''.  
¬  not 

¬ returns true (1) if the operand is false (0), and false (0) if the operand is true (1). In most computer languages negate or not is represented by the exclamation mark '!'.  
⊖  exclusive or 

⊖ returns true (1) if one but not both operands are true (1), otherwise it returns false (0). In most computer languages, exclusive or is implemented as a function call.  
Table 1: Truth table for boolean operators. 
The three binary operators are
commutative:
A∧B = B∧A
A∨B = B∨A
A⊖B = B⊖A.
The three binary operators are
associative:
a ∧ (b ∧ c) = (a ∧ b) ∧ c
a ∨ (b ∨ c) = (a ∨ b) ∨ c
a ⊖ (b ⊖ c) = (a ⊖ b) ⊖ c.
The 'and' and 'or' operators are mutually distributive:
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  Y 
Z  X 
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