Mathematical Logical Reasoning Notes | EduRev

: Mathematical Logical Reasoning Notes | EduRev

 Page 1


14.1 Overview
If an object is either black or white, and if it is not black, then logic leads us to the
conclusion that it must be white. Observe that logical reasoning from the given hypotheses
can not reveal what “black” or “white” mean, or why an object can not be both.
Infact, logic is the study of general patterns of reasoning, without reference to particular
meaning or context.
14.1.1  Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:  No sentence can be called a statement if
(i) It is an exclamation
(ii) It is an order or request
(iii) It is a question
(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.
Example 1
(i) The sentence
‘New Delhi is in India; is true. So it is a statement.
(ii) The sentence
“Every rectangle is a square” is false. So it is a statement.
(iii) The sentence
“Close the door” can not be assigned true or false (Infact, it is a command). So
it can not be called a statement.
(iv) The sentence
Chapter 14
MATHEMATICAL REASONING
Page 2


14.1 Overview
If an object is either black or white, and if it is not black, then logic leads us to the
conclusion that it must be white. Observe that logical reasoning from the given hypotheses
can not reveal what “black” or “white” mean, or why an object can not be both.
Infact, logic is the study of general patterns of reasoning, without reference to particular
meaning or context.
14.1.1  Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:  No sentence can be called a statement if
(i) It is an exclamation
(ii) It is an order or request
(iii) It is a question
(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.
Example 1
(i) The sentence
‘New Delhi is in India; is true. So it is a statement.
(ii) The sentence
“Every rectangle is a square” is false. So it is a statement.
(iii) The sentence
“Close the door” can not be assigned true or false (Infact, it is a command). So
it can not be called a statement.
(iv) The sentence
Chapter 14
MATHEMATICAL REASONING
“How old are you?” can not be assigned true or false (In fact, it is a question).
So it is not a statement.
(v) The truth or falsity of the sentence
“x is a natural number” depends on the value of x. So it is not considered as a
statement. However, in some books it is called an open statement.
Note: Truth and falisity of a statement is called its truth value.
14.1.2  Simple statements A statement is called simple if it can not be broken down
into two or more statements.
Example 2  The statements
“2 is an even number”,
“A square has all its sides equal” and
“ Chandigarh is the capital of Haryana” are all simple statements.
14.1.3  Compound statements  A compound statement is the one which is made up of
two or more simple statements.
Example 3 The statement
“11 is both an odd and prime number” can be broken into two statements
“11 is an odd number” and “11 is a prime number” so it is a compound statement.
Note: The simple statements which constitutes a compound statement are called
component statements.
14.1.4 Basic logical connectives  There are many ways of combining simple
statements to form new statements. The words which combine or change simple
statements to form new statements or compound statements are called Connectives.
The basic connectives (logical) conjunction corresponds to the English word ‘and’;
disjunction corresponds to the word ‘or’; and negation corresponds to the word
‘not’.
Throughout we use the symbol ‘?’ to denote conjunction; ‘?’ to denote disjunction and
the symbol ‘~’ to denote negation.
Note: Negation is called a connective although it does not combine two or more
statements. In fact, it only modifies a statement.
14.1.5  Conjunction  If two simple statements p and q are connected by the word
‘and’, then the resulting compound statement “p and q” is called a conjunction of p
and q and is written in symbolic form as “p ? q”.
MATHEMATICAL REASONING    247
Page 3


14.1 Overview
If an object is either black or white, and if it is not black, then logic leads us to the
conclusion that it must be white. Observe that logical reasoning from the given hypotheses
can not reveal what “black” or “white” mean, or why an object can not be both.
Infact, logic is the study of general patterns of reasoning, without reference to particular
meaning or context.
14.1.1  Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:  No sentence can be called a statement if
(i) It is an exclamation
(ii) It is an order or request
(iii) It is a question
(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.
Example 1
(i) The sentence
‘New Delhi is in India; is true. So it is a statement.
(ii) The sentence
“Every rectangle is a square” is false. So it is a statement.
(iii) The sentence
“Close the door” can not be assigned true or false (Infact, it is a command). So
it can not be called a statement.
(iv) The sentence
Chapter 14
MATHEMATICAL REASONING
“How old are you?” can not be assigned true or false (In fact, it is a question).
So it is not a statement.
(v) The truth or falsity of the sentence
“x is a natural number” depends on the value of x. So it is not considered as a
statement. However, in some books it is called an open statement.
Note: Truth and falisity of a statement is called its truth value.
14.1.2  Simple statements A statement is called simple if it can not be broken down
into two or more statements.
Example 2  The statements
“2 is an even number”,
“A square has all its sides equal” and
“ Chandigarh is the capital of Haryana” are all simple statements.
14.1.3  Compound statements  A compound statement is the one which is made up of
two or more simple statements.
Example 3 The statement
“11 is both an odd and prime number” can be broken into two statements
“11 is an odd number” and “11 is a prime number” so it is a compound statement.
Note: The simple statements which constitutes a compound statement are called
component statements.
14.1.4 Basic logical connectives  There are many ways of combining simple
statements to form new statements. The words which combine or change simple
statements to form new statements or compound statements are called Connectives.
The basic connectives (logical) conjunction corresponds to the English word ‘and’;
disjunction corresponds to the word ‘or’; and negation corresponds to the word
‘not’.
Throughout we use the symbol ‘?’ to denote conjunction; ‘?’ to denote disjunction and
the symbol ‘~’ to denote negation.
Note: Negation is called a connective although it does not combine two or more
statements. In fact, it only modifies a statement.
14.1.5  Conjunction  If two simple statements p and q are connected by the word
‘and’, then the resulting compound statement “p and q” is called a conjunction of p
and q and is written in symbolic form as “p ? q”.
MATHEMATICAL REASONING    247
248    EXEMPLAR PROBLEMS – MATHEMA TICS
Example 4  Form the conjunction of the following simple statements:
p : Dinesh is a boy.
q : Nagma is a girl.
Solution  The conjunction of the statement p and q is given by
p ? q : Dinesh is a boy and Nagma is a girl.
Example 5  Translate the following statement into symbolic form
“Jack and Jill went up the hill.”
Solution  The given statement can be rewritten as
“Jack went up the hill and Jill went up the hill”
Let p : Jack went up the hill   and   q : Jill went up the hill.
Then the given statement in symbolic form is p ? q.
Regarding the truth value of the conjunction p ? q of two simple statements p and q,
we have
(D
1
) : The statement p ? q has the truth value T (true) whenever both p and q
have the truth value T.
(D
2
) : The statement p ? q has the truth value F (false) whenever either p or q
or both have the truth value F.
Example 6  Write the truth value of each of the following four statements:
(i) Delhi is in India and 2 + 3 = 6.
(ii) Delhi is in India and 2 + 3 = 5.
(iii) Delhi is in Nepal and 2 + 3 = 5.
(iv) Delhi is in Nepal and 2 + 3 = 6.
Solution  In view of (D
1
) and (D
2
) above, we observe that statement (i) has the truth
value F as the truth value of the statement “2 + 3 = 6” is F. Also, statement (ii) has the
truth value T as both the statement “Delhi is in India” and “2 + 3 = 5” has the truth
value T .
Similarly, the truth value of both the statements (iii) and (iv) is F.
14.1.6  Disjunction  If two simple statements p and q are connected by the word
‘or’, then the resulting compound statement “p or q” is called disjunction of p and q
and is written in symbolic form as “p ? q”.
Example 7  Form the disjunction of the following simple statements:
p : The sun shines.
q : It rains.
Page 4


14.1 Overview
If an object is either black or white, and if it is not black, then logic leads us to the
conclusion that it must be white. Observe that logical reasoning from the given hypotheses
can not reveal what “black” or “white” mean, or why an object can not be both.
Infact, logic is the study of general patterns of reasoning, without reference to particular
meaning or context.
14.1.1  Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:  No sentence can be called a statement if
(i) It is an exclamation
(ii) It is an order or request
(iii) It is a question
(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.
Example 1
(i) The sentence
‘New Delhi is in India; is true. So it is a statement.
(ii) The sentence
“Every rectangle is a square” is false. So it is a statement.
(iii) The sentence
“Close the door” can not be assigned true or false (Infact, it is a command). So
it can not be called a statement.
(iv) The sentence
Chapter 14
MATHEMATICAL REASONING
“How old are you?” can not be assigned true or false (In fact, it is a question).
So it is not a statement.
(v) The truth or falsity of the sentence
“x is a natural number” depends on the value of x. So it is not considered as a
statement. However, in some books it is called an open statement.
Note: Truth and falisity of a statement is called its truth value.
14.1.2  Simple statements A statement is called simple if it can not be broken down
into two or more statements.
Example 2  The statements
“2 is an even number”,
“A square has all its sides equal” and
“ Chandigarh is the capital of Haryana” are all simple statements.
14.1.3  Compound statements  A compound statement is the one which is made up of
two or more simple statements.
Example 3 The statement
“11 is both an odd and prime number” can be broken into two statements
“11 is an odd number” and “11 is a prime number” so it is a compound statement.
Note: The simple statements which constitutes a compound statement are called
component statements.
14.1.4 Basic logical connectives  There are many ways of combining simple
statements to form new statements. The words which combine or change simple
statements to form new statements or compound statements are called Connectives.
The basic connectives (logical) conjunction corresponds to the English word ‘and’;
disjunction corresponds to the word ‘or’; and negation corresponds to the word
‘not’.
Throughout we use the symbol ‘?’ to denote conjunction; ‘?’ to denote disjunction and
the symbol ‘~’ to denote negation.
Note: Negation is called a connective although it does not combine two or more
statements. In fact, it only modifies a statement.
14.1.5  Conjunction  If two simple statements p and q are connected by the word
‘and’, then the resulting compound statement “p and q” is called a conjunction of p
and q and is written in symbolic form as “p ? q”.
MATHEMATICAL REASONING    247
248    EXEMPLAR PROBLEMS – MATHEMA TICS
Example 4  Form the conjunction of the following simple statements:
p : Dinesh is a boy.
q : Nagma is a girl.
Solution  The conjunction of the statement p and q is given by
p ? q : Dinesh is a boy and Nagma is a girl.
Example 5  Translate the following statement into symbolic form
“Jack and Jill went up the hill.”
Solution  The given statement can be rewritten as
“Jack went up the hill and Jill went up the hill”
Let p : Jack went up the hill   and   q : Jill went up the hill.
Then the given statement in symbolic form is p ? q.
Regarding the truth value of the conjunction p ? q of two simple statements p and q,
we have
(D
1
) : The statement p ? q has the truth value T (true) whenever both p and q
have the truth value T.
(D
2
) : The statement p ? q has the truth value F (false) whenever either p or q
or both have the truth value F.
Example 6  Write the truth value of each of the following four statements:
(i) Delhi is in India and 2 + 3 = 6.
(ii) Delhi is in India and 2 + 3 = 5.
(iii) Delhi is in Nepal and 2 + 3 = 5.
(iv) Delhi is in Nepal and 2 + 3 = 6.
Solution  In view of (D
1
) and (D
2
) above, we observe that statement (i) has the truth
value F as the truth value of the statement “2 + 3 = 6” is F. Also, statement (ii) has the
truth value T as both the statement “Delhi is in India” and “2 + 3 = 5” has the truth
value T .
Similarly, the truth value of both the statements (iii) and (iv) is F.
14.1.6  Disjunction  If two simple statements p and q are connected by the word
‘or’, then the resulting compound statement “p or q” is called disjunction of p and q
and is written in symbolic form as “p ? q”.
Example 7  Form the disjunction of the following simple statements:
p : The sun shines.
q : It rains.
MATHEMATICAL REASONING    249
Solution  The disjunction of the statements p and q is given by
p ? q : The sun shines or it rains.
Regarding the truth value of the disjunction p ? q of two simple statements p and q, we
have
(D
3
) : The statement p ? q has the truth value F whenever both p and q have
the truth value F.
(D
4
) : The statement p ? q has the truth value T whenever either p or q or both
have the truth value T.
Example 8  Write the truth value of each of the following statements:
(i) India is in Asia or 2 + 2 = 4.
(ii) India is in Asia or 2 + 2 = 5.
(iii) India is in Europe or 2 + 2 = 4.
(iv) India is in Europe or 2 + 2 = 5.
Solution In view of (D
3
) and (D
4
) above, we observe that only the last statement has
the truth value F as both the sub-statements “India is in Europe” and “2 + 2 = 5” have
the truth value F. The remaining statements (i) to (iii) have the truth value T as at least
one of the sub-statements of these statements has the truth value T.
14.1.7 Negation  An assertion that a statement fails or denial of a statement is called
the negation of the statement. The negation of a statement is generally formed by
introducing the word “not” at some proper place in the statement or by prefixing the
statement with “It is not the case that” or It is false that”.
The negation of a statement p in symbolic form is written as “~ p”.
Example 9  Write the negation of the statement
p : New Delhi is a city.
Solution  The negation of  p is given by
~ p : New Delhi is not a city
or ~ p : It is not the case that New Delhi is a city.
or ~ p : It is false that New Delhi is a city.
Regarding the truth value of the negation ~ p of a statement p, we have
(D
5
) : ~ p has truth value T whenever  p has truth value F.
(D
6
) : ~ p has truth value F whenever p has truth value T.
Page 5


14.1 Overview
If an object is either black or white, and if it is not black, then logic leads us to the
conclusion that it must be white. Observe that logical reasoning from the given hypotheses
can not reveal what “black” or “white” mean, or why an object can not be both.
Infact, logic is the study of general patterns of reasoning, without reference to particular
meaning or context.
14.1.1  Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:  No sentence can be called a statement if
(i) It is an exclamation
(ii) It is an order or request
(iii) It is a question
(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.
(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.
(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.
Example 1
(i) The sentence
‘New Delhi is in India; is true. So it is a statement.
(ii) The sentence
“Every rectangle is a square” is false. So it is a statement.
(iii) The sentence
“Close the door” can not be assigned true or false (Infact, it is a command). So
it can not be called a statement.
(iv) The sentence
Chapter 14
MATHEMATICAL REASONING
“How old are you?” can not be assigned true or false (In fact, it is a question).
So it is not a statement.
(v) The truth or falsity of the sentence
“x is a natural number” depends on the value of x. So it is not considered as a
statement. However, in some books it is called an open statement.
Note: Truth and falisity of a statement is called its truth value.
14.1.2  Simple statements A statement is called simple if it can not be broken down
into two or more statements.
Example 2  The statements
“2 is an even number”,
“A square has all its sides equal” and
“ Chandigarh is the capital of Haryana” are all simple statements.
14.1.3  Compound statements  A compound statement is the one which is made up of
two or more simple statements.
Example 3 The statement
“11 is both an odd and prime number” can be broken into two statements
“11 is an odd number” and “11 is a prime number” so it is a compound statement.
Note: The simple statements which constitutes a compound statement are called
component statements.
14.1.4 Basic logical connectives  There are many ways of combining simple
statements to form new statements. The words which combine or change simple
statements to form new statements or compound statements are called Connectives.
The basic connectives (logical) conjunction corresponds to the English word ‘and’;
disjunction corresponds to the word ‘or’; and negation corresponds to the word
‘not’.
Throughout we use the symbol ‘?’ to denote conjunction; ‘?’ to denote disjunction and
the symbol ‘~’ to denote negation.
Note: Negation is called a connective although it does not combine two or more
statements. In fact, it only modifies a statement.
14.1.5  Conjunction  If two simple statements p and q are connected by the word
‘and’, then the resulting compound statement “p and q” is called a conjunction of p
and q and is written in symbolic form as “p ? q”.
MATHEMATICAL REASONING    247
248    EXEMPLAR PROBLEMS – MATHEMA TICS
Example 4  Form the conjunction of the following simple statements:
p : Dinesh is a boy.
q : Nagma is a girl.
Solution  The conjunction of the statement p and q is given by
p ? q : Dinesh is a boy and Nagma is a girl.
Example 5  Translate the following statement into symbolic form
“Jack and Jill went up the hill.”
Solution  The given statement can be rewritten as
“Jack went up the hill and Jill went up the hill”
Let p : Jack went up the hill   and   q : Jill went up the hill.
Then the given statement in symbolic form is p ? q.
Regarding the truth value of the conjunction p ? q of two simple statements p and q,
we have
(D
1
) : The statement p ? q has the truth value T (true) whenever both p and q
have the truth value T.
(D
2
) : The statement p ? q has the truth value F (false) whenever either p or q
or both have the truth value F.
Example 6  Write the truth value of each of the following four statements:
(i) Delhi is in India and 2 + 3 = 6.
(ii) Delhi is in India and 2 + 3 = 5.
(iii) Delhi is in Nepal and 2 + 3 = 5.
(iv) Delhi is in Nepal and 2 + 3 = 6.
Solution  In view of (D
1
) and (D
2
) above, we observe that statement (i) has the truth
value F as the truth value of the statement “2 + 3 = 6” is F. Also, statement (ii) has the
truth value T as both the statement “Delhi is in India” and “2 + 3 = 5” has the truth
value T .
Similarly, the truth value of both the statements (iii) and (iv) is F.
14.1.6  Disjunction  If two simple statements p and q are connected by the word
‘or’, then the resulting compound statement “p or q” is called disjunction of p and q
and is written in symbolic form as “p ? q”.
Example 7  Form the disjunction of the following simple statements:
p : The sun shines.
q : It rains.
MATHEMATICAL REASONING    249
Solution  The disjunction of the statements p and q is given by
p ? q : The sun shines or it rains.
Regarding the truth value of the disjunction p ? q of two simple statements p and q, we
have
(D
3
) : The statement p ? q has the truth value F whenever both p and q have
the truth value F.
(D
4
) : The statement p ? q has the truth value T whenever either p or q or both
have the truth value T.
Example 8  Write the truth value of each of the following statements:
(i) India is in Asia or 2 + 2 = 4.
(ii) India is in Asia or 2 + 2 = 5.
(iii) India is in Europe or 2 + 2 = 4.
(iv) India is in Europe or 2 + 2 = 5.
Solution In view of (D
3
) and (D
4
) above, we observe that only the last statement has
the truth value F as both the sub-statements “India is in Europe” and “2 + 2 = 5” have
the truth value F. The remaining statements (i) to (iii) have the truth value T as at least
one of the sub-statements of these statements has the truth value T.
14.1.7 Negation  An assertion that a statement fails or denial of a statement is called
the negation of the statement. The negation of a statement is generally formed by
introducing the word “not” at some proper place in the statement or by prefixing the
statement with “It is not the case that” or It is false that”.
The negation of a statement p in symbolic form is written as “~ p”.
Example 9  Write the negation of the statement
p : New Delhi is a city.
Solution  The negation of  p is given by
~ p : New Delhi is not a city
or ~ p : It is not the case that New Delhi is a city.
or ~ p : It is false that New Delhi is a city.
Regarding the truth value of the negation ~ p of a statement p, we have
(D
5
) : ~ p has truth value T whenever  p has truth value F.
(D
6
) : ~ p has truth value F whenever p has truth value T.
250    EXEMPLAR PROBLEMS – MATHEMA TICS
Example 10  Write the truth value of the negation of each of the following statements:
(i) p : Every square is a rectangle.
(ii) q : The earth is a star.
(iii) r : 2 + 3 < 4
Solution  In view of (D
5
) and (D
6
), we observe that the truth value of ~p is F as the truth
value of p is T. Similarly, the truth value of both ~q and ~r is T as the truth value of both
statements q and r is F.
14.1.8  Negation of compound statements
14.1.9  Negation of conjunction Recall that a conjunction p ? q consists of two
component statements  p and q both of which exist simultaneously. Therefore, the
negation of the conjunction would mean the negation of at least one of the two component
statements. Thus, we have
(D
7
) : The negation of a conjunction p ? q is the disjunction of the negation of
p and the negation of q. Equivalently, we write
~ (p ? q) =  ~ p  ? ~ q
Example 11 Write the negation of each of the following conjunctions:
(a) Paris is in France and London is in England.
(b) 2 + 3 = 5 and 8 < 10.
Solution
(a) Write p : Paris is in France and q : London is in England.
Then, the conjunction in (a) is given by p ? q.
Now ~ p : Paris is not in France, and
~ q : London is not in England.
Therefore, using (D
7
), negation of  p ? q is given by
~ ( p ? q) = Paris is not in France or London is not in England.
(b) Write p : 2 + 3 = 5 and q : 8 < 10.
Then the conjunction in (b) is given by p ? q.
Now ~ p : 2 + 3 ? 5 and ~ q : 8 ? ? 10.
Then, using (D
7
), negation of p ? q is given by
– ( p ? q) = (2 + 3 ? 5 ) or (8 ? ?
10)
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