Introduction To Logical Operations

INTRODUCTION :
Mathematics is an exact science. Every statement in Mathematics must be precise. Also there can’t be Mathematics without proofs and each proof needs proper reasoning. Proper reasoning involves logic. The dictionary meaning of ‘Logic’ is the science of reasoning. The rules of logic gives precise meaning to mathematic statements. These rules are used to distinguished between valid & invalid mathematical arguments.

In addition to its importance in mathematical reasoning, logic has numerous applications in computer science to verify the correctness of programs & to prove the theorems in natural & physical sciences to draw conclusion from experiments, in social sciences & in our daily lives to solve a multitude of problems.

The area of logic that deals with propositions is called the propositional calculus or propositional logic. The mathematical approach to logic was first discussed by British mathematician George Boole; hence the mathematical logic is also called as Boolean logic.

PROPOSITIONS (OR STATEMENTS): A proposition (or a statement) is a declarative sentence that is either true or false, but not both. A proposition (or a statement) is a declarative sentence which is either true or false but not both. Imperative, exclamatory, interrogative or open statements are not statements in logic. Mathematical identities are considered to be statements.

Example 1 : For Example consider, the following sentences.
i) The earth is round.
ii) 4 3 = 7
iii) London is in Denmark
iv) Do your homework
v) Where are you going?
vi) 2 4 = 8
vii) 15 < 4
viii) The square of 4 is 18.
ix) x 1 = 2
x) May God Bless you!

All of them are propositions except iv), v), ix) & x) sentences i), ii) are true, where as iii), iv), vii) & viii) are false. Sentence iv) is command hence not proposition. Is a question so not a statement. ix) Is a declarative sentence but not a statement, since it is true or false depending on the value of x. x) is a exclamatory sentence and so it is not statement.

Mathematical identities are considered to be statements.
Statements which are imperative, exclamatory, interrogative or open are not statements in logic.

Compound statements : Many propositions are composites that is, composed of subpropositions and various connectives discussed subsequently. Such composite propositions are called compound propositions. A proposition is said to be primitive if it can not be broken down into simpler propositions, that is, if it is not composite.

Example 2 : Consider, for example following sentences.
(a) “The sum is shining today and it is cold”
(b) “Juilee is intelligent or studies every night.”

Also the propositions in Example 1 are primitive propositions.