Mathematical
Reasoning
In mathematical language, there are two kinds of reasoning—inductive
and deductive. Here, we will discuss some fundamentals of deductive
reasoning.
Statement (Proposition)
A statement is an assertive sentence which is either true or false but
not both. Statements are denoted by the small letters i.e. p, q , r ... etc.
e.g. p : A triangle has four sides.
Note
(i) A true statement is known as a valid statement and a false statement
is known as an invalid statement.
(ii) Imperative, exclamatory, interrogative, optative sentences are not
statements.
1. Simple Statement
A statement which cannot be broken into two or more statements is
called a simple statement.
e.g. p : 2 is a real number.
2. Open Statement
A sentence which contains one or more variable such that when certain
values are given to the variable it becomes a statement, is called an
open statement.
e.g. p : ‘He is a great man’ is an open statement because in this
statement, he can be replaced by any person.
, 3. Compound Statement
If two or more simple statements are combined by the use of words
such as ‘and’, ‘or’, ‘if... then, ‘if and only if ’, then the resulting
statement is called a compound statement.
e.g. Roses are red and sky is blue.
Note Individual statements of a compound statement are called component
statements.
Elementary Logical Connectives or
Logical Operators
(i) Negation A statement which is formed by changing the truth
value of a given statement by using the word like ‘no’, ‘not' is
called negation of given statement. If p is a statement, then
negation of p is denoted by ~ p.
(ii) Conjunction A compound statement formed by two simple
statements p and q using connective ‘and’ is called the
conjunction of p and q and it is represented by p ∧ q.
(iii) Disjunction A compound statement formed by two simple
statements p and q using connectives ‘or’ is called the
disjunction of p and q and it is represented by p ∨ q.
(iv) Conditional Statement (Implication) Two simple
statements p and q connected by the phrase, if L then, is called
conditional statement of p L q and it is denoted by p ⇒ q.
(v) Biconditional Statement (Bi-implication) The two simple
statements p and q connected by the phrase, ‘if and only if’ is
called biconditional statement. It is denoted by p ⇔ q.
Truth Value and Truth Table
A statement can be either ‘true’ or ‘false’ which is called truth value of
a statement and it is represented by the symbols T and F, respectively.
A truth table is a summary of truth values of the compound
statement for all possible truth values of its component statements.
Logical Equivalent Statements
Two compound statements say, S1 ( p, q , r ) and S 2( p, q , r ,.... ), are said to
be logically equivalent if they have the same truth values for all
logically possibilities. If statements S1 and S 2 are logically equivalent,
then we write
S1( p, q , r... ) = S 2( p, q , r ,... )
Reasoning
In mathematical language, there are two kinds of reasoning—inductive
and deductive. Here, we will discuss some fundamentals of deductive
reasoning.
Statement (Proposition)
A statement is an assertive sentence which is either true or false but
not both. Statements are denoted by the small letters i.e. p, q , r ... etc.
e.g. p : A triangle has four sides.
Note
(i) A true statement is known as a valid statement and a false statement
is known as an invalid statement.
(ii) Imperative, exclamatory, interrogative, optative sentences are not
statements.
1. Simple Statement
A statement which cannot be broken into two or more statements is
called a simple statement.
e.g. p : 2 is a real number.
2. Open Statement
A sentence which contains one or more variable such that when certain
values are given to the variable it becomes a statement, is called an
open statement.
e.g. p : ‘He is a great man’ is an open statement because in this
statement, he can be replaced by any person.
, 3. Compound Statement
If two or more simple statements are combined by the use of words
such as ‘and’, ‘or’, ‘if... then, ‘if and only if ’, then the resulting
statement is called a compound statement.
e.g. Roses are red and sky is blue.
Note Individual statements of a compound statement are called component
statements.
Elementary Logical Connectives or
Logical Operators
(i) Negation A statement which is formed by changing the truth
value of a given statement by using the word like ‘no’, ‘not' is
called negation of given statement. If p is a statement, then
negation of p is denoted by ~ p.
(ii) Conjunction A compound statement formed by two simple
statements p and q using connective ‘and’ is called the
conjunction of p and q and it is represented by p ∧ q.
(iii) Disjunction A compound statement formed by two simple
statements p and q using connectives ‘or’ is called the
disjunction of p and q and it is represented by p ∨ q.
(iv) Conditional Statement (Implication) Two simple
statements p and q connected by the phrase, if L then, is called
conditional statement of p L q and it is denoted by p ⇒ q.
(v) Biconditional Statement (Bi-implication) The two simple
statements p and q connected by the phrase, ‘if and only if’ is
called biconditional statement. It is denoted by p ⇔ q.
Truth Value and Truth Table
A statement can be either ‘true’ or ‘false’ which is called truth value of
a statement and it is represented by the symbols T and F, respectively.
A truth table is a summary of truth values of the compound
statement for all possible truth values of its component statements.
Logical Equivalent Statements
Two compound statements say, S1 ( p, q , r ) and S 2( p, q , r ,.... ), are said to
be logically equivalent if they have the same truth values for all
logically possibilities. If statements S1 and S 2 are logically equivalent,
then we write
S1( p, q , r... ) = S 2( p, q , r ,... )
