1. SUBTOPIC 3 : METHOD OF PROOF.
From correct statements to an incorrect conclusion. Some other forms of argument
(“fallacies”) can lead from true statements to an incorrect conclusion.
Def: An axiom is a statement that is assuming to be true, or in the case of a mathematical system,
is used to specify the system.
Def: A mathematical argument is a list of statements. Its last statement is called the conclusion.
Def: A logical rule of inference is a method that depends on logic alone for deriving a new
statement from a set of other statements.
Def: A mathematical rule of inference is a method for deriving a new statement that may depend
on inferential rules of a mathematical system as well as on logic.
3.1 Tautology and contradiction.
Mathematical induction is a method of mathematical proof typically used to establish that
a given statement is true for all natural numbers (positive integers). It is done by proving that the
first statement in the infinite sequence of statements is true, and then proving that if any one
statement in the infinite sequence of statements is true, then so is the next one.
Tautology:
A proposition that is always true for all possible value of its propositional variables.
Example of a Tautology
The compound proposition p ˅ ¬p is a tautology because it is always true.
P ¬p p ˅ ¬p
T F T
F T T
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2. Two propositional expressions P and Q are logically equivalent, if and only if P ↔ Q is a
tautology. We write P ≡ Q or P ↔ Q.
A compound proposition that is always false is called a contradiction. A proposition that
is neither a tautology nor contradiction is called a contingency.
Note that the symbols ≡ and ↔ are not logical connectives.
Contradiction:
A proposition that is always false for all possible values of its propositional variables.
Example of a Contradiction
The compound proposition p ^ ¬p is a contradiction because it is always false.
P ¬p p ˅ ¬p
T F F
F T F
A proof by contradiction is based on the idea that if an assumption leads to an absurdity
or to something that could not possibly be true, then the assumption must be false.
Usage of tautologies and contradictions - in proving the validity of arguments; for
rewriting expressions using only the basic connectives.
Contingency:
A proposition that can either be true or false depending on the truth values of its
propositional variables.
A compound proposition that is neither a tautology nor a contradiction is called a
contingency
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3. A contingency table is a table of counts. A two-dimensional contingency table is formed
by classifying subjects by two variables. One variable determines the row categories; the other
variable defines the column categories. The combinations of row and column categories are
called cells. Examples include classifying subjects by sex (male/female) and smoking status
(current/former/never) or by "type of prenatal care" and "whether the birth required a neonatal
ICU" (yes/no). For the mathematician, a two-dimensional contingency table with r rows and c
columns is the set {xi j: i =1... r; j=1... c}.
Propositional form Propositions that are substitution instances of that form
Tautologous Logically true
Contingent Contingently true, Contingently false
Contradictory Logically false
3.2 Argument and rules of inference
Arguments based on tautology represent universally correct methods of reasoning. The
validity of the arguments depends only on the form of the statements involved and not on the
truth values of the variables.
Definition:
An argument is a sequence of propositions written
:
∴ q
Or , ,… / ∴ q.
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4. The symbol ∴ is read “therefore.” The propositions , ,… are called the hypotheses
(or premises) or the proposition q is called the conclusion. The argument is valid provide that if
the proposition are all true, then q must also be true; otherwise, the argument is invalid (or a
fallacy).
Example:
Determine whether the argument
p→q
p
∴q
Is valid
[First solution] We construct a truth table for all the propositions involved.
P q p→q p q
T T T T T
T F F T F
F T T F T
F F T F F
Example:
A Logical Argument
If I dance all night, then I get tired.
I danced all night.
Therefore I got tired.
Logical representation of underlying variables:
p: I dance all night. q: I get tired.
Logical analysis of argument:
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5. p→q premise 1
p premise 2
q Conclusion
Def: A form of logical argument is valid if whenever every premise is true, the conclusion is also
true. A form of argument that is not valid is called a fallacy.
We shall see why the argument above is valid. This form of argument is called modus ponens.
The argument is used extensively and is known as the modus ponens rule of inference or
law of detachment. Several useful rules of inference for propositions, which may be verified
using truth table.
Rule of inference Name
p→q Modus ponens
p
∴q
p→q Modus tollens
⌐q
∴ ⌐p
Addition
p
∴p˅q
Simplification
p˅q
∴p
p Conjunction
q
∴p˅q
p→q Hypothetical syllogism
q→r
∴p→r
p˅q Disjunctive syllogism
⌐p
∴q
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6. Example1:
Represent the argument.
The bug is either in module 17 or in module 81
The bug is a numerical error
Module 81 has no numerical error
___________________________________________
∴ the bug is in module 17.
Given the beginning of this section symbolically and show that it is valid.
If we let
p : the bug is in module 17.
q : the bug is in module 81.
r : the bug is numerical error.
The argument maybe written
p ˅q
r
r → ⌐q
∴p
From r → ⌐q and r, we may use modus ponens to conclude ⌐q. From r ˅ q and ⌐q, we
may use the disjunctive syllogism to conclude p. Thus the conclusion p follows from the
hypotheses and the argument is valid.
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7. Direct proof
This method is based on Modus Ponens,
[(p ⇒ q) ˅ p ]⇒ q
Virtually all mathematical theorems are composed of implication of the type,
(
The are called the hypothesis or premise, and q is called conclusion. To prove a
theorem means to show the implication is a tautology. If all the are true, the q must be also
true.
To directly establish the implication p q by showing if p is true, then q is true. Note
that we do not need to show the cases when p is false! All we need to do is to show:
If p is true, q has to be true.
To prove a proposition in the form p q, we begin by assuming that p is true and then
show that q must be true.
Example:
An even number is of the form 2n where n is an integer, whereas an odd number is 2n + 1. Prove
that if x is an odd integer then x2 is also odd.
Solution:
Let p: x is odd, and q: x2 is odd. We want to prove p q.
Start: p: x is odd
x = 2n + 1 for some integer n
x2 = (2n + 1)2
x2 = 4n2 + 4n + 1
x2 = 2(2n2 + 2n) + 1
x2 = 2m + 1, where m = (2n2 + 2n) is an integer
x2 is odd
q
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8. Example:
Alt Proof of Disjunctive Syllogism: by a chain of inferences.
p ˅q Premise 1
q ˅p commutatively of _
¬¬q ˅ p Double negation law
¬q p A B, ¬A B
¬p Premise 2
¬¬q Modus tollens
q Conclusion by negation
Example: A theorem
The sum of two even numbers x and y is even.
Proof:
There exist numbers m and n such that x = 2m and y = 2n (by def of “even”).
Then x + y = 2m + 2n (by substitution).
= 2(m + n) (by left distributive)
This is even, by the definition of evenness.
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9. Indirect proof
Definition:
An indirect proof uses rules of inference on the negation of the conclusion and on some of
the premises to derive the negation of a premise. This result is called a contradiction.
Contradiction: to prove a conditional proposition p ⇒ q by contradiction, we first assume
that the hypothesis p is true and the conclusion is false (p˅ ~ q). We then use the steps from the
proof of ~q ⇒ ~p to show that ~p is true. This leads to a contradiction (p˅ ~ p which complete
),
the proof.
Example: A theorem
If is odd, then so is x.
Proof: Assume that x is even (negation of conclusion).
Say x = 2n (definition of even).
Then = (substitution)
= 2n · 2n (definition of exponentiation)
= 2 · 2n2 (commutatively of multiplication.)
Which is an even number (definition of even)
This contradicts the premise that is odd.
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10. EXERCISE:
1.
Assume is rational. Then , where and b are relatively prime integers and
.
2. is an irrational number.
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11. ANSWER:
1. Proving this directly (via constructive proof) would probably be very difficult--if not
impossible. However, by contradiction we have a fairly simple proof.
Proposition 2.3.1.
Proof: Assume is rational. Then where and are relatively prime integers and
. So
But since is even, must be even as well, since the square of an odd number is also odd.
Then we have , or
so .
The same argument can now be applied to to find . However, this contradicts the
original assumption that a and b are relatively prime, and the above is impossible. Therefore, we
must conclude that is irrational.
Of course, we now note that there was nothing in this proof that was special about 2, except the
fact that it was prime. That's what allowed us to say that was even since we knew that was
even. Note that this would not work for 4 (mainly because ) because does
not imply that
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12. 2. Proof. Assume that is a rational number. Then, = a/b for two positive integers a
and b. Assume that a and b have no common factors so that the fraction a/b is an
irreducible fraction. By squaring both sides of = a/b, we deduce 2 = / . Therefore
=2
which implies that a2 is even. From Proposition 2, we conclude that a is even, i.e., a = 2k
for some integer k. Substitute a = 2k in equation (1) to get
We conclude that b2 is even which implies that b is even. We have derived that
both a and b are even but this a contradiction since we assumed that the fraction a/b was
irreducible. Therefore, is an irrational number.
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