Introduction
Theorems and Proofs
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Theorems and Proofs
102
Previous Lecture
Equivalent predicates
Equivalent quantified statements
Quantifiers and conjunction/disjunction
Quantifiers and logic connectives
Equivalence and multiple quantifiers
Axioms and theorems
Discrete Mathematics – Theorems and Proofs
103
Proving Theorems
To prove theorems we use rules of inference.
Usually implicitly
In axiomatic theories it is done explicitly:
Specify axioms
Specify rules of inference
Elementary geometry is an axiomatic theory.
Axioms are Euclid’s postulates
Discrete Mathematics – Theorems and Proofs
104
Proving Theorems
We know rules of inference to reason about propositional
statements.
What about predicates and quantified statements?
The simplest method is the
method of exhaustion:
To prove that
2200
x P(x),
just verify that
P(a)
is true for all values
a
from the universe.
To prove that
5
x P(x),
by checking all the values in the universe
find a value
a
such that
P(a)
is true
``Every car in lot C is red’’
``There is a blue car in lot C’’
Discrete Mathematics – Theorems and Proofs
105
Rule of Universal Specification
Reconsider
the argument
Every man is mortal.
Socrates is a man.
∴
Socrates is mortal
In symbolic form it looks like
2200
x (P(x)
→
Q(x))
P(Socrates)
∴
Q(Socrates)
where
P(x)
stands for
x
is a man, and
Q(x)
stands for
x
is mortal
Discrete Mathematics – Theorems and Proofs
106
Rule of Universal Specification (cntd)
If an open statement becomes true for all values of the universe,
then it is true for each specific individual value from that universe
2200
x P(x)
∴
P(c)
Example
Premises:
2200
x (P(x)
→
Q(x)),
P(Socrates)
Step
Reason
1.
2200
x (P(x)
→
Q(x)),
premise
2.
P(Socrates)
→
Q(Socrates),
rule of universal specification
3.
P(Socrates)
premise
4.
Q(Socrates)
modus ponens
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107
Rule of Universal Generalization
Let us prove a theorem:
If
2x – 6 = 0
then
x = 3.
Proof
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 Spring '08
 PEARCE
 Logic

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