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

178 Chapter 5 Discrete mathematics
5.2.3 Intersection
Given setsAandB, a new set which contains the elements common to bothAandBis
called theintersectionofAandB, written as
A∩B={x:x∈Aandx∈B}
In Example 5.2, we see thatA ∩B = {3,5}, that is 3 ∈A ∩B and 5 ∈A ∩B. If the set
A ∩BhasnoelementswesaythesetsAandBaredisjointandwriteA ∩B = ◦/,where
◦/denotes theempty set.
Aset with no elements iscalled an empty set and isdenoted by ◦/.
IfA∩B=◦/, thenAandBare disjoint sets.
5.2.4 Union
Given two setsAandB, the set which contains all the elements ofAand those ofBis
called theunionofAandB, written as
A∪B={x:x∈Aorx∈Borboth}
InExample5.2,A∪B = {1,2,3,4,5,7,9}.Wenotethatalthoughtheelements3and5
are common toboth setsthey are listedonly once.
5.2.5 Subsets
If all the members of a setAare also members of a setBwe sayAis a subset ofBand
writeA ⊂B. We have already metanumber ofsubsets.Convince yourself that
N⊂ZandZ⊂R
Example5.3 IfM representsthesetofallcapacitorsmanufacturedbymachineM,andM f
represents
the faulty capacitors made by machine M, then clearlyM f
⊂M.
5.2.6 Complement
If we are given a well-defined universal set E and a setAwithA ⊂ E, then the set of
membersof EthatarenotinAiscalledthecomplementofAandiswrittenasA.Clearly
A ∪A = E. There areno members inthe setA∩A, thatisA ∩A = ◦/.
Example5.4 A company has a number of machines which manufacture thyristors. We consider only
two machines, M and N. A small proportion made by each is faulty. Denoting the sets
offaultythyristorsmadebyMandNbyM f
andN f
,respectively,depictthissituationon
a Venn diagram.Describethe setsM f
∪N f
andM∪N.
Solution Let E be the universal set of all thyristors manufactured by the company. The Venn
diagram is shown in Figure 5.2. Note in particular that M ∩N = ◦/. There can be no
thyristorsintheintersectionsinceifathyristorismadebymachineMitcannotbemade