Why are Quantified Statements Important The logical structure of quantified statements provides a basis for the construction and validation of proofs Consider the quantified statements For every positive number ID: 377672
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Slide1
A Quick Look at Quantified StatementsSlide2
Why
are Quantified Statements Important?
The logical structure of quantified statements provides a basis for the construction and validation of
proofs.Slide3
Consider the quantified statements…
“For every positive number
b
, there exists a positive number
a
such that
a
<
b
.
“There exists a positive number
a
such that for every positive number
b
,
a
<
b
.”
Are the statements true or false?Slide4
A mathematical interpretation of a quantified statement relies on the quantified structure of the statement.
By convention, a quantified statement is interpreted as it is written from left to right.Slide5
Mathematical Convention for Interpreting Statements “for all… there exists…” (AE)
{
b
0
,
b
1
,
b
2
,
b3, b4, …}
{a0, a1, a2, a3, a4, …}Slide6
Mathematical Convention for Interpreting Statements “there exists… for all…” (EA)
{
b
0
,
b
1
,
b
2
,
b3, b4, …}{
a0, a1, a2, a3, a4, …}Slide7
Consider the quantified statements…
“For every positive number
b
, there exists a positive number
a
such that
a
<
b
.
The statement is true. For any positive number
b, we can find a positive number a less than it; for instance, let a=b/2. “There exists a positive number
a such that for every positive number b, a<b.”The statement is false. There is no positive number a smaller than every other positive number b; for instance, let b
=a/2