CSCE 235
Handout
10: Proofs
February 4, 2004
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Terms |
Definitions |
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Theorem |
A theorem is a statement that can be shown to be true. Theorems are sometimes called propositions, facts, or results. |
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Proof |
We show a theorem is true with a sequence of statements that form an argument, called proof. To construct proofs, methods are needed to derive new statements from old ones. |
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Axioms or Postulates |
These statements are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, and previously proved theorems, used in a proof. |
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Rules of Inference |
These are the means used to draw conclusions from other assertions, tying together the steps of a proof. |
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Fallacies |
These are common forms of incorrect reasoning |
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Lemma |
A lemma (plural lemmas or lemmata) is a simple theorem used in the proof of other theorems. Complicated proofs are usually easier to understand when they are proved using a series of lemmas, where each lemma is proved individually. |
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Corollary |
A corollary is a proposition that can be established directly from a theorem that has been proved. |
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Conjecture |
A conjecture is a statement whose truth value is unknown. When a proof of a conjecture is found, the conjecture becomes a theorem. Many times conjectures are shown to be false, so they are not theorems. |
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Fallacy of affirming the conclusion. |
The proposition |
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Fallacy of denying the
hypothesis. |
The proposition |
Methods
of Proving Theorems
Direct Proofs:
In which the hypotheses H1, … Hn are shown to imply the conclusion C.
.
For example, the
implication 
can be proved by
showing that if p is true, then q must also be true. This shows that the combination p
true and q false never occurs.
Vacuous proofs
An implication 
is sometimes said to
be vacuously true if p is false.
This is because we have decreed that 
is true whenever p is
false and so, in this case, the truth of 
tells us
nothing. A vacuous proof is a
proof of an implication 
in which it is shown
that p is false.
Trivial Proofs
An implication 
is sometimes said to
be trivially true if q is true.
This is because, in this case, the truth value of p is irrelevant. A trivial proof of 
is one in which q is
shown to be true without any reference to p.
Indirect Proofs:
Proof of the Contrapositive
Since the implication 
is equivalent to the
contrapositive 
, the implication can be proved by showing that its
contrapositive is true.
Indirect Proofs:
Proof by Contradiction
Sometimes a direct proof of a statement seems hopeless. We simply do not know how to begin. In this case, we can sometimes make progress by assuming that the negation of A is true. If this assumption leads to a statement which is obviously false (an “absurdity”) or to a statement which contradicts something else, then we will have shown that “not A” is false. So A must be true.
Indirect
Proofs: Proof by Cases
Sometimes, a direct theorem (argument) is made simpler by breaking it into a number of cases, with each leading to the desired conclusions.
To prove:
we do
Indirect
Proofs: Proofs of Equivalence
To prove a theorem that
is a biconditional, that is, one that is a statement of the form 
where p
and q are propositions, the tautology 
can be used.
Existence Proofs
A theorem of this type
is a proposition of the form 
, where P is a predicate. Such a proof is called an existence proof. Sometimes an existence proof of 
can be given by
finding an element a such that 
is true. Such an existence proof is called constructive. There are also nonconstructive proofs
in which we prove that 
is true in some other
way (such as contradiction).
Uniqueness Proofs
Some theorems assert the existence of a unique element with a particular property. To prove a statement of this type, we need to show that an element with this property exists and that no other element has this property. The two parts:
Existence: We show that an element x with the desired property exists.
Uniqueness: We show that if y is not x, then y does not have the desired property.