mathematical logic

Let us Recoll
o The converse, inverse ond contropositive of the implicotiorl
p+ q ore:
Converse : Q)P
Inverse : -P+-Q
Contropositive : - q-+- P
o Quontifiers qnd quontified Ststements : Look ot the following
stotements:
p : "There exists on even prime number in the set of nqturol numbers"'
q : "A11 noturol numbers ore positive"'
Eoch of them osserts o condition for some or oll objects in o collection.
.'there exists" qnd "for oll" ore colled quontifiers. "There exists"
Words
qll" is
is colled existentiol quontifier ond is denoted by symbol :. "For
colled universol quontifier ond is denoted by V ' Stqtements involving
quontifiers ore colled quontified ststements. Every quontified stotement
p the
corresponds to o collection ond o condition' In stotement
collection is 'the set of noturol numbers' ond the condition is 'being
even prime'.
Whot is the condition in the stotement q ?
A stqtement quontified by universol quontifier V is true if o11 objects
in the collection sotisfy the condition. And it is folse if ot leost one
object in the collection does not sotisfy the condition.
A stotement quontified by existentiol quontifier : is true if ot leost
one object in the collection sotisfy the condition. And it is folse if no
object in the collection sotisfy the condition'
Idempotent Low p^p=p' pvp=p
Commutqtive Low pYq=clvp pna-_!_!_L
Associotive Low p (q n r) = (P " q) A r :- P A q A r
p ^Y (q v r) = (P " q) , !:J_, q u:
Distributive Low pn(q"r)=(Pnq)v(P!r)
p
"G;i = (p v q) n (P:)
De Morgon's Low * n q) : - p Y -Q, -@ v q) = -P n -q
Identity Low WF,pv F=p,pv T=T
Complement Low p,.-p=F,Pv-P=T