Mathematical Logic

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Introduction

     The study of logic through the use of mathematical symbols is called Mathematical Logic. Mathematical logic is also known as Symbolic Logic or Boolean Logic.

Proposition or statement and Truth Value of a statement

     A declarative sentence which is true or false but not both is called a proposition. Truth value is the truthfulness of statement. If a statement is true its truth value is T, if a statement is false then its truth value is F.

Use of Venn diagrams in Logic

     Venn diagrams are used very frequently on problems of “set theory”. Venn diagrams can also be used for deciding the truthfulness of statements.

Compound statements and Truth table

     If two or more statements are combined by the use of words like 'AND', 'OR', 'IF AND ONLY IF ', then the resulting statement is called a compound statement. A table is indicating the truth values of one or more statements is called a truth table.

Logical Equivalence and Duality

     Two compound propositions p and q are said to be logically equivalent, if their truth values are same for each different combinations of the truth values of the components involved in them. If p and q are logically equivalent, then it is represented by p º q.

Application of Logic in solving simple problems

     An argument is a statement which assets that a given set of n compound statements p₁,p₂,..........p_n yield another compound statement Q.

Summary

     -    A sentence is a statement if it is either true or false, but not both.
     -    The truth and falsity of a statement is its truth value.
   -   A truth table indicates the truth values of a number of statements and their compound statements in a compact form.
     -    If there are n statements, then there are 2ⁿ rows in the truth table.