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

176 Chapter 5 Discrete mathematics
Theobjectsmaybenumbers,letters,daysoftheweek,or,infact,anythingunderdiscussion.Onewayofdescribingasetistolistthewholecollectionofmembersorelements
and enclose them inbraces { }.Consider the following examples.
A = {1,0}
B = {off, on}
C = {high, low}
D = {0,1,2,3,4,5,6,7,8,9}
the set ofbinary digits, one and zero
the set of possible statesof a two-statesystem
the set of effective voltage levels inadigital electronic
circuit
the set ofdigits used inthe decimal system
Notice that we usually use a capital letter to represent a set. To state that a particular
object belongs to a particular set we use the symbol ∈ which means ‘is a member of’.
So, forexample, wecan write
off∈B
3∈D
Likewise, /∈means ‘is notamember of’so that
low /∈B
5/∈A
are sensiblestatements.
Listingmembersofasetisfinewhentherearerelativelyfewbutisuselessifweare
dealingwithverylargesets.Clearly,wecouldnotpossiblywritedownallthemembers
of the set of whole numbers because there are an infinite number. To assist us special
symbols have been introduced tostand for some commonly used sets. These are
N the set of non-negative whole numbers, 0,1,2,3,...
N + the set of positive whole numbers, 1,2,3,...
Z the set of whole numbers, positive, negative and zero,
...−3,−2,−1,0,1,2,3...
R the set of all real numbers
R + the set of positive real numbers
R − the set of negative real numbers
Q the set of rational numbers
Note thatareal number isany number inthe interval (−∞,∞).
Another way of defining a set is to give a rule by which all members can be found.
Consider the following notation:
A={x:x∈Randx<2}
Thisreads‘Aisthesetofvalues ofxsuchthatxisamember ofthesetofrealnumbers
andxis less than 2’. ThusAcorresponds to the interval (−∞,2). Using this notation
other setscan be defined.
Note that
R + ={x:x∈Randx>0}
R − ={x:x∈Randx<0}