CSCE 235
Handout
4: Quantifiers in Predicate Logic
January 29, 2003
Quantifiers
The Universal
quantifier:
Definition
1: The universal quantification of
is the proposition
“
is true for all values of x in the universe of
discourse.”
The universe of discourse is the domain.
The notation 
denotes the universal
quantification of 
. Here 
is called the universal
quantifier. The proposition 
is read as “for all x
” or “for every x 
”.
To show that a
statement of the form 
is false, where 
is a propositional
function, we need only find one value of x in the universe
of discourse for which 
is false. Such a value of x is called a counterexample
to the statement 
.
The Existential
quantifier:
With existential
quantification, we form a proposition that is true iff 
is true for at least
one value of x in the universe of discourse.
Definition
2: The existential
quantification of 
is the proposition
“There
exists an element x in the universe of discourse such that 
is true.”
The universe of discourse is the domain.
The notation 
denotes the
existential quantification of 
. Here 
is called the existential
quantifier. The proposition 
is read as “There is
an x such that 
” or “There is at least one x such that 
”. Or “For some x 
”/
To show that a
statement of the form 
is true, where 
is a propositional
function, we need only find one value of x in the universe
of discourse for which 
is true. Such a value of x is called a proof
to the statement 
.
|
Statement |
When True? |
When False? |
|
|
|
There is an x for
which |
|
|
There is an x
for which |
|
Negations
|
Statement |
When True? |
When False? |
|
|
|
There is a pair x,
y for which |
|
|
For every x,
there is a y for which |
There is an x such
that |
|
|
There is an x for
which |
For every x
there is a y for which |
|
|
There is a pair x,
y for which |
|
