World Library  
Flag as Inappropriate
Email this Article

Inclusion (Boolean algebra)

Article Id: WHEBN0040153832
Reproduction Date:

Title: Inclusion (Boolean algebra)  
Author: World Heritage Encyclopedia
Language: English
Subject: Subset, Boolean algebra
Publisher: World Heritage Encyclopedia

Inclusion (Boolean algebra)

In Boolean algebra (structure), the inclusion relation a\le b is defined as ab'=0 and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.

The inclusion relation a can be expressed in many ways:

  • a
  • ab'=0
  • a'+b=1
  • b'
  • a+b=b
  • ab=a

The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.

Some useful properties of the inclusion relation are:

  • a\le a+b
  • ab\le a

The inclusion relation may be used to define Boolean intervals such that a\le x\le b A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.


  • Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 52
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.