Logic
The meaning of logic is the science of
reasoning. In mathematics, we deal with various theorem and formulae. The
different procedure used in giving the proofs of various theorem and formulae
are based on reasoning. The study of such procedures based on reasoning is
known as logic. In logic, we use symbols for the words, statements and their
relations to get the required result. Hence, logic is known as mathematical
logic or symbolic logic.
Statements:
A declarative sentence,
which is either true or false but not both true or false at the same time is
known as statement.
Example of statements
(i) Water is
essential for health.(true)
(ii) 2 + 4 = 6.(true)
(iii) 2×3
= 5.(false)
(iv) Triangle has four
sides.(false)
(v) 2 is an irrational
number.(false)
The following
sentences are not statements:
i) Open
the door.( Imperative)
ii) How
old are you?( introgative)
iii) How beautiful our country is ! (exclamatory)
iv) x
+ 2 = 4.(Open sentence)
v) …..
is a son of Khagendra.(Open sentence)
The above sentences
are neither true nor false.
Imperative,
introgative, exclamatory and open sentences are not statements.
Simple statement
A statement which declares
only one thing is known as a simple statement. In simple statement, sentence
cannot be divided into two or more sentences.
Compound statements
A
combination of two or more simple statements is known as compound statement.
Compound statement can be divided into two or more simple statements.
Truth
value
The truth or falsity of an statement is known
as its truth value.
The truth or falsity of
the simple statement depends upon the truth or falsity of the given statement.
But in compound statement, its truth value depends not only on the truth or
falsity of the of the compound statements but also the connectives with which
the component statements are combined.
Truth tables
The table presenting the
truth values of the component statements together with the truth values of
their compound staements, is known as the truth table.
Logical
connectives
The compound statements
are formed from the simple statements by using words or phrase like
"and", "or", if ... then, it and only if, either ... or,
neither ... nor etc. and they are called logical connectives.
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