Logic

 

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.