Download Rules of Inference and Methods of Proof

Rules of Inference and Methods
of Proof
Mariam Kreydem
[email protected]
Office No.: 152-C
Building No.: 07
Year: 2016
Mariam Kreydem, 2016
What is a Mathematical Argument?
A mathematical argument is a sequence of propositions 1 that are ordered
in steps such that the appearance of the later step depends on the truth value
of the previous ones.
The body of any argument can be divided into two parts:
- Premises: all but the final proposition in the argument.
- Conclusion: the final proposition of the argument.
So, to show the validity of any argument we show that the conclusion of
the argument must follow from the truth of the preceding statements which
are obtained by using premises. Equivalently,
An argument is valid if and only if it is impossible for all the premises to be
true and the conclusion to be false.
Different Way to Build a Logical Argument
To deduce new statements from statements we already have, we use rules
of inference which are templates for constructing valid arguments by establishing the truth of their statements. In what follows is a list of the most
famous rules of inference that are used to build an argument.
1- Modus Ponens (the law of detachment): which has the following
form
p
p
→
q
−− − − − − −
∴q
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In logic, a proposition is a sentence which is either true or false. Compare this definition with that of the proof’s part.
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This rule can be summarized as: p implies q; but p is asserted to be true,
therefore q must be true.
Example 1. Let p be the statement ”If it snows today,”
and q be the assertion that ”then we will go skiing.”
Note that if the conditional statement ”p → q” is true provided that its
premise p is also true, then the conclusion is q.
2- Modus Tollens: which has the following form
¬q
p
→
q
−− − − − − −
∴ ¬p
This rule means that if ”p → q” is true but the conclusion q is not true,
then the premise p must be false.
3- Hypothetical Syllogism: which has the following form
p
→
q
→
−− − − −
∴p
→
q
r
−−
r
This rule is also called also the chain argument and the chain rule.
Example 2. If I do not wake up,
then I cannot go to work.
If I cannot go to work,
then I will not get paid.
Therefore, if I do not wake up, then I will not get paid.
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4- Disjunctive Syllogism: which has the following form
p
∨
q
¬p
−− − − − − −
∴q
This rule says that if either p or q is true and p is false, then q is true.
Example 3. It is either red or blue.
It is not blue.
Therefore, it is red.
5- Addition: which has the following form
p
−− − − − − −
∴p
∨
q
This rule says that it is enough for the truth of a preposition with connection ” ∨ ” is to have one of p or q to be true, say p as in our case.
6- Simplification: which has the following form
p
∧
q
−− − − − − −
∴p
7- Conjunction: which has the following form
p
q
−− − − − − −
∴p
∧
q
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8- Resolution: which has the following form
p
∨
¬p
∨
−− − − −
∴q
∨
q
r
−−
r
Examples of Establishing Arguments Using
the Rules of Inference
Example 4. Assume we have the following statements: ”It is not sunny this
afternoon, and it is colder than yesterday. We will go swimming only if it is
sunny. If we do not go swimming, then we will take a canoe trip. If we take
a canoe trip, then we will be home by sunset”. To prove the validity of this
argument using the rules of inference, let
p: It is sunny this afternoon,
q: It is colder than yesterday.
r: We will go swimming.
s: We will take a canoe trip.
t: We will be home by sunset.
Steps
¬p ∧ q
¬p
r→p
¬r
¬r → s
s
s→t
t
Reason
premise
simplification using rule (6)
premise
modus tollens
premise
modus ponens
premise
modus ponens
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Example 5. Assume we have the following statements: ”If you send me an
e-mail message, then I will finish writing the program. If you do not send me
an e-mail message, then I will go to sleep early. If I go to sleep early, then I
will wake up feeling refreshed”. To prove the validity of this argument using
the rules of inference, let
p:
q:
r:
s:
If you send me an e-mail message,
I will finish writing the program.
I will go to sleep early.
I will wake up feeling refreshed.
Steps
p→q
¬q → ¬p
¬p → r
¬q → r
r→s
¬q → s
Reason
premise
contrapositive
premise
hypothetical syllogism
premise
hypothetical syllogism
Now, let us have a look at the following example, and see what goes wrong
with it.
Example 6. Determine whether the argument given here is valid and determine whether its conclusion must be true because of the validity of the
argument.
√
√
√
”If 2 > 32 , then ( 2)2 > ( 32 )2 . We know that 2 > 32 . Consequently,
√
( 2)2 = 2 > ( 32 )2 = 94 .”
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√
Solution. Let p be the proposition ” 2 > 32 ” and q the proposition
”2 > ( 32 )2 ”. The premises of the argument are p → q and p is its premise,
and q is its conclusion. This argument is valid because it is constructed by
using
modus ponens, a valid argument form. However, one of its premises,
√
2 > 23 , is false. Consequently, we cannot conclude that the conclusion is
true. Furthermore, note that the conclusion of this argument is false, because
2 < 49 .
Remark 1. A valid argument can lead to an incorrect conclusion if one or
more of its premises is false.
Remark (1) leads to the following concept: fallacy.
Fallacies
The word fallacy can be defined as a mistake in reasoning used to support or refute an argument. A fallacious argument may seem to prove its
conclusion but, in fact, it does not.
In what follows are the most famous types of fallacies:
1- Fallacy of Affirming the Conclusion: which takes the form
p
→
q
q
−− − − − − −
∴p
This argument is invalid, since if the conclusion is true this does not mean
that the premise p is always true.
Exercise. Give a mathematical example of such a fallacy.
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2- Fallacy of Denying the Hypothesis: which have the form
p
→
q
¬p
−− − − − − −
∴ ¬q
The mistake in this argument is that the invalidity of premises does not
necessarily leads to the invalidity of the conclusion.
Exercise. Give a mathematical example for this fallacy.
What is a Proof ?
A proof is a demonstration that something is true. In mathematics, we
prove the validity of mathematical statements called results by verifying
the truth of the argument used for this purpose.
What to prove?
We prove different results with different names. We call these results:
theorems, propositions, lemma, corollary, ..., etc.
A theorem is a mathematical assertion that can be shown to be true.
Many theorems assert that a property holds for all elements in a domain.
In other words, it has a form of the conditional statement:
∀x(P (x) → Q(x)).
(1)
A lemma is a theorem used to prove other theorems.
A corollary is a proposition that can be proved as a consequence of a
theorem that has just been proved.
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A proposition is a less important theorem.
Different types of proofs
To prove a result of a form (1), we have to recall first the truth table of
the conditional proposition:
p → q.
Recall the following table 2 :
Based on this table we the following different types of proof.
1- Vacuous Proof : a proof that p → q is true based on the fact that p
is false.
So, from the table; this will not affect the truth of the result since if p is
false, and q is true or false, then in both cases we get p → q is true.
Example 7. The empty set is a subset of any set.
2
See Kenneth Rosen’s book.
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Solution Let S be any set. Let p be the proposition ”x ∈ ∅” and q be
the assertion ”x ∈ S”. Since p is false, then no matter what values q takes,
the proposition
p→q
is true. The poof of the result is completed.
2- Trivial Proof : a proof that p → q is true based on the fact that q is
true.
In other words, most of time we can decide the truth of q without the
need of p.
Example 8. If a and b are integers and a ≥ b, then an ≥ bn . Show that the
statement is true at n = 0.
Solution. If n = 0, then the conclusion becomes 1 = a0 ≥ b0 = 1.
3- Direct Proof : a proof that p → q is true that proceeds by showing
that q must be true when p is true.
Example 9. If n is an odd integer, then n2 is odd.
Solution. Exercise.
4- Proof by Contraposition: a proof that p → q is true that proceeds
by showing that p must be false when q is false. In other words,
¬q → ¬p.
The reason why this type of proof is valid is the fact the
p → q ≡ ¬q → ¬p.
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Example 10. If n is an integer, and 3n + 2 is odd, then n is odd.
Solution. Exercise.
5-Proof by Contradiction: a proof that p → q is true based on the
fact that
¬q ∧ p → F ≡ p → q.
This means that we suppose by contradiction the conclusion q is false and
when put it together with the premise p, we get ¬p. So, at the end we have
p ∧ ¬p
which is impossible. Another way for the contradiction to occur is to get the
impossible case q ∧ ¬q as follows:
Example 11. Prove that
√
2 is irrational.
Solution. Exercise.
6-Without Loss of Generality (WLOG): an assumption in a proof
that makes it possible to prove a theorem by reducing the number of cases
to consider in the proof.
Example 12. If x and y are integers, and both xy and x + y are even, then
both x and y are even.
Solution. First, suppose that x and y are not both even. With out loss
of generality, suppose that x is not even. The rest of the proof is left for the
reader to do. 7-Counterexample: we need to find an element x such that P (x) is
false. By this way, we show that ∀xP (x) is false.
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Example 13. Every positive integer is the sum of the sequence of two integers.
Solution. We can show that this statement is false by using a counterexample: the number 3 can not be written as a sequence of two integers. We
leave to the reader to check this fact.
8-Uniqueness Proof : a proof that there is exactly one element satisfying a specified property. To do so, two steps are needed:
First: the existence of such an element, that is, there exists x with the
desired property P (x).
Second: uniqueness, that is, if for every y such that x 6= y, then y does
not have the property. Symbolically,
∃x(P (x) ∧ ∀(y 6= x → ¬P (y))).
Equivalently, if x and y both have the desired property, then x = y.
Example 14. If x and y are real numbers, and x 6= 0, then there is a unique
real number r such that ar + b = 0.
Solution. Exercise.
9-Forward and Backward Proof : A forward approach is simply replacing premises’ statements with other statements that follow from them,
until we reach the conclusion statement. Unfortunately, forward proofs
are often difficult to find. A variation on the forward proof is the forward/backward proof. In this case, we proceed from both ends and work
toward the middle, and go as far as we can. Then we start with and go as
far backwards as we can. Hopefully, at some point, we arrive at a common
point. From this, we can then rewrite the steps as a forward proof.
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Reference
[1] Rosen, K. H. (2012). Discrete Mathematics and Its Application. Seventh edition. The McGraw-Hill Companies, New York, the United State of
America.
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