Mathematics for Computer Science

(Frankie) #1

Chapter 3 Logical Formulas46


that look like the familiar ones for multiplication of numbers:


AANDB! BANDA commutativity ofAND (3.7)
.AANDB/ANDC! AAND.BANDC/ associativity ofAND (3.8)
TANDA! A identity forAND (3.9)
FANDA! F zero forAND (3.10)

Three axioms that don’t directly correspond to number properties are


AANDA !A idempotence forAND (3.11)
AANDA !F contradiction forAND (3.12)
NOT.A/ !A double negation (3.13)
(3.14)

It is associativity (3.8) that justifies writingAANDBANDC without specifying
whether it is parenthesized asAAND.BANDC/or.AANDB/ANDC. That’s
because both ways of inserting parentheses yield equivalent formulas.
There are a corresponding set of equivalences forORwhich we won’t bother to
list, except for theORrule corresponding to contradiction forAND:


AORA !T validity forOR (3.15)
(3.16)

There is also a familiar rule connectingANDandOR:

AAND.BORC/! .AANDB/OR.AANDC/ distributivity ofANDoverOR


(3.17)

Finally, there areDeMorgan’s Lawswhich explain how to distributeNOT’s over
AND’s andOR’s:


NOT.AANDB/ !AORB DeMorgan forAND (3.18)
NOT.AORB/ !AANDB DeMorgan forOR (3.19)

All these axioms can be verified easily with truth tables.
These axioms are all that’s needed to convert any formula to a disjunctive normal
form. We can illustrate how they work by applying them to turn the negation of
formula (3.4), namely,


NOT..AANDB/OR.AANDC//: (3.20)
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