Download Chapter 5: Section 5-1 Mathematical Logic

Chapter 5: Section 5-1
Mathematical Logic
D. S. Malik
Creighton University, Omaha, NE
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Mathematical Logic
Intuitively, logic is the discipline that considers the methods of
reasoning.
It provides the rules and techniques for determining whether an
argument is valid or not.
In everyday life, we use reasoning to prove di¤erent points.
For example, to prove to our parents that we passed an exam, we
might show the test and the score.
Similarly, in mathematics, mathematical logic (or logic) is used to
prove results.
To be speci…c, in mathematics, we use logic or logical reasoning to
prove theorems.
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Statements (Proposition)
De…nition
A statement or a proposition is a declarative sentence that is either true
or false, but not both.
Example
Consider the following sentences.
(i) 4pis an integer.
(ii) 5 is an integer.
(iii) Washington, DC, is the capital of the USA.
Each of these sentences is a declarative sentence. The sentence (i) is true,
(ii) is not true, and (iii) is true. Hence, these are examples of statements.
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We use lowercase letters, with or without subscripts, such as p, q, and r ,
to denote sentences or statements. For example, we can write
p : 4pis an integer.
q : 5 is an integer.
r : Washington, DC, is the capital of the U.S.A.
Example
Consider the following sentences:
p : 5 is greater than 3.
q : 7 is an even integer.
Both p and q are declarative sentences. Thus, p and q are statements.
Moreover, p is true and q is false.
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Example
Consider the following sentences.
p : Will you go?
q : Shall we enjoy the lovely weather?
Here the sentences p and q are not declarative sentences, so these are not
statements.
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A statement is a declarative sentence that can be classi…ed as true or
false, but not both.
One of the values ‘truth’or ‘false’that is assigned to a statement is
called its truth value.
We abbreviate ‘truth’to T or 1 and ‘false’to F or 0.
If a statement p is true, we say that the (logical) truth value of p is
true and write p is T (or p is 1); otherwise, we say the (logical)
truth value of p is false and write p is F (or p is 0).
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Compound Statements
Consider the following statements.
1
I will not go to the basketball game today.
2
I paid my bill in the morning and I went to the gym in the evening.
3
You can go to Chicago or you can go to Paris.
4
If I win the competition, then I will win the scholarship money.
5
Study hard and get good grades, or lose the scholarship.
These are examples of compound statements. Moreover, not, and, or, and,
if ... then are called logical connectives or simply connectives.
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Example
(i) Consider the statement “4 is a prime number”. The negation of this
statement is “4 is not a prime number”. Hence, “4 is not a prime
number” is a compound statement.
(iii) You can borrow the car and buy the groceries. This is a compound
statement because it contains the connective and.
(iv) If ABC is a right-angled triangle, then one of its angle is 90o . This is a
compound statement because it contains the connective if ... then.
(v) I invested in the Smith and Barney company. This is not a compound
statement. The word and is part of the name of the company.
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Negation
De…nition
Let p be a statement. The negation of p, written
obtained by negating the statement p.
p, is the statement
The truth value of p and p are opposite.
The symbol s is called ‘not’. We read s p as ‘not p’.
If p is a statement, then its negation is formed by writing “it is not
the case that p”. For example, if
p : 2 is positive,
then
s p : it is not the case that 2 is positive.
Sometimes, s p is also written as follows:
s p : 2 is not positive.
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By the de…nition of the negation of a statement p, the truth value of
s p is opposite to the truth value of p, i.e., if p is T , then s p is F
and if p is F , then s p is T . We record this in a table, called a truth
table, as follows:
p sp
T
F
F
T
Example
Consider the statement
The Rockie Mountains are in Colorado.
The negation of this statement is
The Rockie Mountains are not in Colorado.
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Conjunction
Consider the following statements:
p : 2 is an even integer.
q : 7 divides 14.
Now consider the sentence
r : 2 is an even integer and 7 divides 14.
Because r is true, r is a statement. Such a statement r is called the
conjunction of p and q.
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Conjunction
De…nition
Let p and q be statements. The conjunction of p and q, written p ^ q, is
the statement formed by joining the statements p and q using the word
‘and’. The statement p ^ q is true if both p and q are true; otherwise
p ^ q is false.
The symbol ^ is called ‘and’. Let
of p ^ q is given by:
p
T
T
F
F
p and q be statements. The truth table
q
T
F
T
F
p^q
T
F
F
F
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Example
(i) Let p : Washington, DC, is the capital of the USA and q : The USA is
in North America. Then p ^ q is the statement:
p ^ q : Washington, DC, is the capital of the USA and
the USA is in North America.
Notice that p ^ q is T .
(ii) Let p : 2 divides 4 and q : 3 is greater than 5. Then p ^ q is the
statement:
p ^ q : 2 divides 4 and 3 is greater than 5.
Because p is T and q is F , it follows that p ^ q is F .
(iii) Let r : 2 divides 4 and s : 2 divides 6. Then r ^ s is the statement:
r ^ s : 2 divides 4 and 2 divides 6.
Because r is T and s is T , it follows that r ^ s is T .
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Consider the statements, r : 2 divides 4 and s : 2 divides 6, in the
previous example. The statement r ^ s is:
r ^ s : 2 divides 4 and 2 divides 6.
Sometimes, the statement r ^ s is written
r ^ s : 2 divides both 4 and 6.
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Disjunction
Given two statements p and q, we can form the statement ‘p or q’by
putting the word ‘or’between the statements such that the statement p or
q is true if at least one of the statements p or q is true. For example,
suppose we have the statements:
p : 2 is an integer.
q : 3 is greater than 5.
Then we can form the statement.
r : 2 is an integer or 3 is greater than 5.
Because p is T , it follows that r is true. The statement r is called the
disjunction of p and q.
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Disjunction
De…nition
Let p and q be statements. The disjunction of p and q, written p _ q, is
the statement formed by putting the statements p and q together using
the word ‘or’. The truth value of the statement p _ q is T if at least one
of the statements p or q is true.
The symbol _ is called ‘or ’. For statements p and q, the truth table of p _
q is given by:
p q p_q
T T
T
T F
T
F T
T
F F
F
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Example
Let p be the statement “You can pay me now” and q be the statement
“You can pay me tomorrow with a 10% penalty”. Then p _ q is the
statement:
You can pay me now or you can pay me tomorrow with a 10% penalty.
Example
Let p : 22 + 33 is an even integer and q : 22 + 33 is an odd integer. Then
p _ q : 22 + 33 is an even integer or 22 + 33 is an odd integer.
Sometimes for better readability, we write p _ q as:
p _ q : Either 22 + 33 is an even integer or 22 + 33 is an odd integer.
or
p _ q : 22 + 33 is an even integer or an odd integer.
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Implication
In every day life, we encounter statements such as
If it is cold, then I will wear a jacket.
If I get a bonus, then I will buy a car.
If I work, then I must pay taxes.
Similarly, in mathematics, we frequently encounter statements such as:
If ABC is a triangle, then \A + \B + \C = 1800 .
If 49301 is divisible by 6, then 49301 is divisible by 3.
In each of these statements, two statements are connected by If ... then to
form a new statement.
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Implication
De…nition
Let p and q be two statements. Then ‘if p then q’is a statement called an
implication or a condition, written p ! q.
The statement p ! q can also be read as
p implies q
or
p is su¢ cient for q
or
q if p
or
q whenever p.
In the implication p ! q, p is called the hypothesis (or antecedent) and
q is called the conclusion (or consequent).
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Let us consider the following implication:
If I pass Monday’s math test, then I will take you out for dinner.
Let p represent the statement “I pass Monday’s math test” and q
represent the statement “I will take you out for dinner.”
Suppose p is true and q is true. Then because I told the truth, the
statement p ! q is true.
Suppose p is true but q is false. In this case, I lied. So the statement
p ! q is false.
Suppose p is false but q is true. In this case, even though I did not
pass the test, but I still took you out for dinner. Note that I did not
say anything about dinner and failing the test. Therefore, I did not
lie. So the statement p ! q is true.
Suppose p is false and q is false. In this case, I did not pass the test
and I did not take you out for dinner. I cannot be held responsible for
not taking you out for dinner because I promised to take you out for
dinner only if I passed the test. Therefore, I did not lie. So the
statement p ! q is true.
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The truth table of the implication p ! q is given by:
p
T
T
F
F
q
T
F
T
F
p!q
T
F
T
T
Example
Let p be the statement “It rains today”, and q be the statement “The
game will be postponed”. Then the statement p ! q is the statement
If it rains today, then the game will be postponed.
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Evaluating Logical Expressions
Consider the statements:
p = 2 is an even integer
q = 6 is a prime number
Then the truth value of p is true (T ) and the truth value of q is false (F ).
Now p, ( p ) _ q, p ! ( q ) are also statements. Suppose that we
want to know the truth values of these statements.
To …nd the truth value of these statements, we substitute the values of the
variables p and q and then use the rules of evaluating the logical
connectives. For example,
p=
Thus, the truth value of
T = F.
p is F .
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Similarly,
(
p) _ q = ( T ) _ F
= F _ F,
= F,
because
T =F
and
p!(
q) = T ! ( F )
= T !T
= T.
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To avoid the use of so many parentheses (and brackets) in a statement
formula we adopt the following conventions:
1. We omit the outer pair of parentheses in a statement. For example,
we write s p for (s p ).
2. If there is a statement of the form (s p ) ! (s q ), then we write it
as s p ! s q.
3. Similarly, we write (s p ) _ (s q ) as s p _ s q, (s p ) ^ (s q ) as
s p ^ s q, s (s p ) as s s p, and so on.
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Precedence of Logical Connectives (Operators)
In a compound statement without parentheses that contains logical
connectives, the logical connectives are evaluated in the following order,
i.e., the precedence of logical connectives is:
s
^
_
!
highest
second highest
third highest
fourth highest
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Example
Let p be a true (T ) statement, q be a false (F ) statement, and r be a
true (T ) statement.
(i)
p _ (s r ^ q ) = T _ (s T ^ F )
= T _ (F ^ F )
= T _F
= T.
(ii)
(s p ^ q ) _ (s q _ r ) =
=
=
=
(s T ^ F ) _ (s F _ T )
(F ^ F ) _ (T _ T )
F _T
T.
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Example
Let p be a true (T ) statement, q be a false (F ) statement, and r be a
true (T ) statement.
(i)
p !s r = T !s T
= T !F
= F.
(ii)
r ! (s p _ s q ) =
=
=
=
T ! (s T _ s F )
T ! (F _ T )
T !T
T.
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Exercise: Write the negation of each of the following statements:
(i) 2 is an odd integer;
(ii) 7 + 5 > 13;
(iii) p
7;
(iv) I am taking the applied mathematics course.
Solution: (i) 2 is not an odd integer;
(ii) 7 + 5
13
(iii) p > 7;
(iv) I am not taking the applied mathematics course.
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Exercise: Let p be true, q be false, and r be true. Find the truth value
of the following statements.
(i) p ^ (q _ r )
(ii) [(p _ q )^ r )]
(iii) (p _ q ) ! ( q _ r )
Solution: (i) p ^ (q _ r ) = T ^ (F _ T ) = T ^ T = T
(ii)
[(p _ q )^ r ] = [(T _ F )^ T ]
=
[(T _ T ) ^ F ] = [T ^ F ] = F = T
(iii)
(p _ q ) ! ( q _ r ) =
=
T ! (T _ T )
= F !T
= T
(T _ F ) ! (
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