082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

5.2 Set theory 179
E
M
N
M f
Nf
Figure5.2
Venn diagram forExample 5.4.
by machine N and vice versa. Thus M and N are disjoint sets. Also note M f
⊂ M,
N f
⊂N.ThesetM f
∪N f
isthesetoffaultythyristorsmanufacturedbyeithermachineM
or N.The setM∪N isthe set of thyristors made by machines other than M or N.
Wehaveseenhowtheoperations ∩, ∪canbeusedtodefinenewsets.Itisnotdifficult
to show that a number of laws hold, most of which are obvious from the inspection of
anappropriateVenn diagram.
5.2.7 Lawsofsetalgebra
For any setsA,B,C and a universal set E, we have the laws in Table 5.1. From these it
ispossible toprove the laws given inTable 5.2.
Table5.1
Thelaws ofset algebra.
}
A∪B=B∪A
Commutative laws
A∩B=B∩A
}
A∪(B∪C)=(A∪B)∪C
Associativelaws
A∩(B∩C)=(A∩B)∩C
}
A∩(B∪C)=(A∩B)∪(A∩C)
Distributivelaws
A∪(B∩C)=(A∪B)∩(A∪C)
}
A ∪◦/ =A
Identity laws
A∩E=A
⎫
A ∪A = E ⎪⎬
A ∩A = ◦/ Complement laws
A =A
⎪⎭
Table5.2
Laws derivable from Table 5.1.
}
A∪(A∩B)=A
Absorption laws
A∩(A∪B)=A
}
(A∩B)∪(A∩B)=A
Minimization laws
(A∪B)∩(A∪B)=A
}
A∪B=A ∩B
De Morgan’slaws
A∩B=A ∪B
5.2.8 Setsandfunctions
Ifwearegiventwosets,AandB,ausefulexerciseistoexaminerelationships,givenby
rules,betweentheelementsofAandtheelementsofB.Forexample,ifA = {0,1,4,9}