Boolean Algebra

Pronunciation: /ˈbu li ən ˈæl dʒə brə/ ?

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.

OperatorNameTruth TableVenn DiagramDescription
and
ABA ∧ B
000
010
100
111
Venn diagram showing AND, conjunction 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
ABA ∨ B
000
011
101
111
Venn diagram showing 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
A¬A
01
10
Venn diagram showing negation ¬ 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
ABA ⊖ B
000
011
101
110
Venn diagram showing 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.

Properties of Operations

Commutative

The three binary operators are commutative:
A∧B = B∧A
A∨B = B∨A
A⊖B = B⊖A.

Associative

The three binary operators are associative:
a ∧ (b ∧ c) = (a ∧ b) ∧ c
a ∨ (b ∨ c) = (a ∨ b) ∨ c
a ⊖ (b ⊖ c) = (a ⊖ b) ⊖ c
.

Distributive

The 'and' and 'or' operators are mutually distributive:
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
.

References

  1. boolean. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2009-03-12). http://www.merriam-webster.com/dictionary/boolean.
  2. Boole, George. The Mathematical Analysis of Logic. Macmillan, Barclay and Macmillan, 1847. (Accessed: 2009-12-24). http://www.archive.org/stream/mathematicalanal00booluoft#page/n6/mode/1up.
  3. Couturat, Louis. The Algebra of Logic. English translation by Lydia Gillingham Robinson, B. A.. Open Court Publishing, 1914. (Accessed: 2009-12-19). http://www.archive.org/stream/algebralogicbylc00coutrich#page/n5/mode/1up.

More Information

  • boolean. Dictionary.com. Random House, Inc. 2009-03-12. http://dictionary.reference.com/browse/boolean.

Cite this article as:


Boolean Algebra. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/booleanalgebra.html.

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2008-06-30: Added Venn diagrams (McAdams, David.)
2008-04-22: Initial version (McAdams, David.)

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