Math 100
Expressing Deduction
Martin H. Weissman
With the language of sets and numbers, one can express many interesting mathematical facts. We have also
learned how to use boolean operators (AND, OR, XOR, and NOT) to combine sentences into compound
sentences.
But to
do
mathematics, one needs some more “connective tissue” to express a deductive relationship. The
most basic example of this is the “If... then... ” sentence structure.
1.
Basic and compound deduction
Many facts in basic algebra are most conveniently expressed using an “If...then...”
statement.
Consider
these examples:
•
If 2
x
+ 3 = 5, then
x
= 1.
•
If
x
is a real number, then
x
2
is nonnegative.
•
If
x
is an even number, then
∃
y
such that 2
y
=
x
.
•
If

x

3

= 2, then (
x
= 5 or
x
= 1).
Try to complete the following to make a
true
“If...then...” sentence:
(1) If..., then
x >
1.
(2) If..., then
x
= 1 or
x
=

1.
(3) If
x
is an even prime number, then...
(4) If
x
2
=
x
, then...
The “If...then...” statement is the most basic expression of
deduction
. It involves a single step of deduction
– one sentence follows deductively from another. We will discuss methods of deduction later – for now, we
are focusing only on the
language
of deduction.
More often than not, mathematics involves a
compound
deduction. One sentence deductively follows another,
which follows another, etc... Here are some templates for multistep deductions:
(1) If 2
x
+ 3 = 5, then 2
x
= 2. It follows that
x
= 1.
(2) If

x

3

= 2, then
x

3 = 2 or
x

3 =

2. Hence,
x
= 5 or
x
= 1.
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 Fall '08
 Weissman
 Logic, Addition, Sets

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