Let U designate a well defined universe and A, B, and C sets within the universe.

Law of double complement      (A')' = A

Note: recall from logic: (Ø (Ø P)   º  P ).  The student should verify these laws have corresponding laws of logic.
 

Contrapositive form of subset      A Í  B   º  B' Í  A'   º  A'  Ê   B'

De Morgan Law (1)                          A'  Ç  B'   º   (A  È  B)'
De Morgan Law (2)                          A'  È  B'  Û (A  Ç  B)'

Law of the Excluded Middle (1)             x Î A Ç  A'    º always false
Law of the Excluded Middle (2)             x Î A È  A'      º always true
Law of the Excluded Middle (3)             A Ç  A'    =  Æ
Law of the Excluded Middle (4)             A È  A'    = U
 

Commutative Law  of  "or"  (1)        A  È  B  º    B  È  A
Commutative Law  of  "and" (2)       A  Ç  B  º    B  Ç  A

Associative Law  of  "or"  (1)           A  È  (B  È  C)    º  (A  È  B)  È  C  º  A  È  B  È  C
Associative Law  of  "and" (2)         (A  Ç  B)  Ç  C º  (A  Ç  B)  Ç  C º  (A  Ç  B)  Ç  C

Distributive Law of "and over or" (1)      A  Ç  (B  È  C)    º  (A  Ç  B)  È  (A  Ç  C)
Distributive Law of "or over and" (2)       A  È  (B  Ç  C)    º  (A  È  B)  Ç  (A  È  C)

Idempotent Law (1)                                 A  È  A  = A
Idempotent Law (2)                                 A  Ç  A  =  A

Law of Addition                                     x Î A   Þ  x Î A È  B

Law of Simplification                              x Î A Ç  B  Þ  x Î A

Modus Ponens                                      x Î A  Ù   A Í  B     Þ  x Π B

Modus Tollens                                     x Ï B  Ù   A Í  B     Þ  x Ï A

Disjunctive Syllogism                             x Î A  È  B  Ù   x Ï B    Þ  x Π A

Hypothetical Syllogism                          A Í  B   Ù   B Í  C    Þ A Í  C
(Transitivity)
 

Identity Law (1)                                                   A  È  U  = U
Identity Law (2)                                                   A  Ç  Æ   =  Æ
Identity Law (3)                                                   A  Ç  U  = A
Identity Law (4)                                                 A  È  Æ   =  A

Complement Law (1)                                                  U'  =  Æ
Complement Law (2)                                                 Æ'   =  U

 "  set A Í U ,  ( Æ Í A )
 
 
 

Statement:                                                        Negation:

 "  x Î U ,  (x Π A   Þ  x Π B)                   $  x Î U ,  (x Π A   Ù    x Ï  B)

 $  x Î U ,  (x Π A   Þ  x Π B)                   "  x Î U ,  (x Π A   Ù    x Ï  B)

A Í  B                                                               A Ë  B
 

Page 100:    S14 and S15 are wrong! Please correct them.

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© 1999, 2000. M. P. M. M. McLoughlin.