1
Existential Quantifiers: The Construction Method
1.1
Welcome
Welcome to the fourth module in our course on proofs,“Existential Quantifiers: The Construction
Method”. Before going on, please do the assigned readings.
1.2
Introduction
Not all mathematical statements are obviously in the form “If
A
, then
B
”.
You will encounter
statements of the form
there is
,
there are
,
there exists
or
for all
,
for each
,
for every
,
for any
. The
first three are all examples of the
existential quantifier
there is
and the final four are all examples
of the
universal quantifier
for all
. The word
existence
is used to make it clear that we are looking
for or looking at a particular mathematical object. The word
universal
is used to make it clear that
we are looking for or looking at a set of objects all of which share some desired behaviour. This
module deals with existential quantifiers. The next module deals with universal quantifiers.
1.3
Basic Structure and Examples
All statements which use existential quantifiers share a basic structure, though some elements of
the structure may be absent, implicit or appear in a different order.
There is an “object” in the “universe of discourse” with a “certain property” such that
“something happens”.
It is very important to be able to identify the object, the universe of discourse, the property and the
something that happens. The expression “such that” (or equivalent words like “for which”) usually
precede the something that happens.
The
universe of discourse
is the set from which the “object” is taken. More formally, it is the
domain of the object.
Here are some examples.
1. There exists a real number
x
so that
f
(
x
) = 0.
Object:
x
Universe of discourse:
R
Certain property:
None specified
Something happens:
f
(
x
) = 0
This is a good point to illustrate the universe of discourse. Suppose in this example we are
interested in the specific function
f
(
x
) =
x
2

2. Then the statement
There exists a real number
x
such that
x
2

2 = 0.
is true since we can find an
x
,
√
2, so that
x
2

2 = 0.
But if we change the universe of discourse to integers,
There exists an integer
x
such that
x
2

2 = 0.
1
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the statement is false because neither of the two roots,
√
2 or

√
2 are integers. So changing
the universe of discourse can change the truth value of the statement. In practice, the universe
of discourse is often not explicitly stated and is inferred from context.
2. There exists an angle
θ
such that sin(
θ
) = 1.
Object:
angle
θ
Universe of discourse:
R
, inferred from context
Certain property:
None specified
Something happens:
sin(
θ
) = 1
In this example, no property is specified and the universe of discourse is implicit. Note also
that there can be many objects, that is many angles
θ
, which satisfy the statement.
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 Spring '08
 ANDREWCHILDS
 Math, Complex number, certain property

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