Set Identities { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies, and introductions to philosophers, and explanations on a number of topics. Accredited homeschooling online at Northgate Academy and Philosophy online tutoring.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WTI offers immigration law course online - fully accredited. ACE credits online at EES.

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Set Identities

Set identities are methods of expressing the same set using the names of sets and set operations. They can be used in the algebra of sets.

Note that in these examples, A, B and C are sets, and U denotes the universal set — that is, the set containing all elements in the domain. ∅ denotes the empty set. Note also that, in these examples, an absolute complement is written AC. It may also be written as A, A′ or ∁(A) in different sources.

Identity Laws

A ∪ ∅ = A

A ∩ U = A

Domination Laws

A ∪ U = U

A ∩ ∅ = ∅

Complement Laws

A ∪ AC = U

A ∩ AC = ∅

Idempotent Laws

A ∪ A = A

A ∩ A = A

Involution or Double Complement Law

(AC)C = A

Absorption Laws

A ∪ (A ∩ B) = A

A ∩ (A ∪ B) = A

Associative Laws

A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C

Communative Laws

A ∪ B = B ∪ A

A ∩ B = B ∩ A

Distributive Laws

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

De Morgan’s Laws

(A ∪ B)C = AC ∩ BC

(A ∩ B)C = AC ∪ BC

Note: The term ‘De Morgan’s Laws’ also refers to rules of inference in logic.

Set Complement Laws

A - B = A ∩ BC