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

186 Chapter 5 Discrete mathematics
Table5.8
Laws ofBoolean 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+0=A
Identity laws
A·1=A
⎫
A +A = 1⎪⎬
A ·A = 0 Complement laws
A =A
⎪⎭
Table5.9
Laws derived from the laws ofTable 5.8.
}
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
DeMorgan’s laws
A·B=A +B
A+1=1
A·0=0
Example5.9 Findthe truth tableforthe BooleanexpressionA +B·C.
Solution WeconstructthetablebynotingthatA,BandC areBooleanvariables;thatis,theycan
takethevalues0or1.Thefirststageintheprocessistoformallpossiblecombinationsof
A,BandC,asshowninTable5.10.ThenwecompletethetablebyformingC,thenB·C
and finallyA +B·C, using the truth tables defined earlier. So, for example, whenever
C=1,C = 0. The complete process is shown in Table 5.11. Work through the table to
ensure you understand how itwas constructed.
Table5.10
The possible
combinations forthree
variables,A,BandC.
A B C
Table5.11
Thetruthtable forA +B·C.
A B C C B·C A+B·C
1 1 1
1 1 0
1 0 1
1 0 0
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1 0 0 1
1 1 0 1 1 1
1 0 1 0 0 1
1 0 0 1 0 1
0 1 1 0 0 0
0 1 0 1 1 1
0 0 1 0 0 0
0 0 0 1 0 0