Mathematical Reasoning

Mathematical Reasoning

INTRODUCTION

Logic is the subject that deals with the methods of reasoning without reference to particular meaning or context. How well the ability to reason can be used depends upon a person's power to put forward his reasoning. The process of reasoning is relevant to deductive mathematics.

Sentence

Logic plays a definitive role in the process of reasoning; we communicate our ideas or thoughts with the help of sentences in particular languages. Following types of sentences are normally used:

i) Assertive Sentence:

A sentence that makes an assertion (declaration) is called an assertive or declarative sentence.

ii) Imperative Sentence:

A sentence that expresses a request or a command is called an imperative sentence.

iii) Exclamatory Sentence:

A sentence that expresses some strong feeling is called an exclamatory sentence.

iv) Interrogative Sentence:

A sentence that asks some question is called an interrogative sentence.

v) Optative Sentence:

A sentence that expresses a wish is called an optative sentence.

Statement:

A statement is an assertive sentence which is either true or false but not both simultaneously.

Note:

A sentence which is both true and false simultaneously is not a statement, rather it is a paradox.

Open Statement:

Let A be a given set. Then a declarative sentence P (n) containing a variable x such that P (a) is true or false for each a A is called an open statement defined on A.


Truth Set:

The set of all those values of the variable in an open statement for which it becomes a true statement is called the truth set of open statement.


Truth Value of a Statement:

If a statement is true, then its truth value is TRUE or T. If it is false then its truth value is FALSE or F.

Logical Variables:

Statements are represented by lower case letters such as p, q, r, ..... etc. These letters are called logical variables.


Simple Statement:

A statement is said to be simple if it cannot be broken into two or more sentences.

Compound Statement:

A statement which is formed by combining two or more simple statements is called a compound statements.


Sub Statements:

Simple statements which on combining form a compound statement are called sub statements or component statements of the compound statements.

Note:

The truth value of compound statement is determined by the truth value of each of its sub statements.



Basic Logical Connectives or Logical Operators

Connectives:

The phrases or words which connect simple statements to form compound statements are called logical connectives or connectives or logical operators.

Conjunction:

Two simple statements can be connected by the word "and" to form a compound statement called the conjunction of the original statements. We use symbol "

" to denote the conjunction.

If p and q are two simple statements then p q denote the conjunction of p and q and it is read as "p and q".


Note:

Symbol has specific meaning corresponding to connectives, although it may also be used with some other meaning.


The truth value of the conjunction p q of two simple statements p and q:

(a) The statement p q has the truth value T whenever both p and q have the truth value T.

(b) The statement p q has the truth value F whenever either p or q or both have the truth value F.


Disjunction or Alternation:

Two statements can be connected by the word "or" to form a compound statement, called the disjunction of the original statements.

If p and q are two simple statements then p q denotes the disjunction of p and q and it is read as "p or q".


The truth value of the disjunction p q:

(a) The statement p q has the truth value F if both p and q have the truth value F.

(b) The statement p q has the truth value T whenever either p or q or both have the truth value T.

Note:

Connective word 'or' is also used with other meaning.


Negation:

An assertion that a statement fails or denial of a statement is called the negation of a statement.

The negation of a statement is generally formed by introducing the word "no" at some proper place in the statement or by prefixing the statement with "it is not the
case that" or "it is false that". The negation of a statement p is denoted by "~p" or " p".

Note:

Negation is called a connective although it does not combine two or more statements.


Negation of Compound Statements:

If p and q are two sub statements.


(a) Negation of Conjunction:

The negation of a conjunction p q is the disjunction of the negation of p and the negation of q.


(b) Negation of Disjunction:

The negation of a disjunction p q is the conjunction of the negation of p and the negation of q.


(c) Negation of a Negation:

Negation is not a connective but a modifier. It only modifies a given statement and applies only to a single simple statement.

Negation of negation of a statement is the statement itself i.e. ~ (~ p) p.

Conditional and Biconditional Statements:

If statements are of the from "if p then q" and "q if and only if p", such statements are called conditional statements.


Implication or Conditional Statements:

Two statements, when connected by the connective phrase 'if ........ then', give a compound statement known as an implication or a conditional statement.

If p and q are two statements then it is denoted by "p q" or "p q" and read as "p implies q".

Here p is called antecedent and q is the consequent.


Converse, Inverse and Contrapositive of an Implication:

If p and q are two statements, then

Implication: p q

Converse: q p

Inverse: ~ p ~ q

Contrapositive ~ q ~ p.


Biconditional or Equivalence Statement:

Two simple statement, connected by the phrase "if and only if" give a biconditional statement. It is the conjunction of the conditional statements, one converse to

the other. If p and q are two statements then biconditional statement is denoted by "p q" or "p q" so that

p q : (p q) (q p)


Negation of Conditional Statement:

i) Negation of Implication:

The negation of an implication p q is the conjunction of p and the negation of q

i.e. ~ (p q) p ~ q.

ii) Negation of Biconditional Statement:

The negation of equivalence p q is disjunction of the negation of implication p q and the negation of implication q p.

Joint Denial:

The phrases "neither .... nor" connecting two simple sentences is called the joint denial, denoted by .

Proposition:

A compound statement having component statement p, q, r, ..... etc. with repetitive use of the connectives , , , and ~ is called a proposition.


Tautologies:

A statement is said to be a tautology if it is true for all logical possibilities. In other words a statement is called tautology if its truth value is T and only T.


Contradictions:

A statement is said to be contradiction if it is false for all logical possibilities. In other words a statement is called contradictions if its truth value is F and only F.


Contingency:

A statement is said to be contingency if it is either true or false depending on truth value of its components.

Note:

The negation of a tautology is a contradiction and vice versa.


Logical Equivalence:

Two statements S₁ (p, q, r, ......) and S₂ (p, q, r, ........) are said to be logically equivalent or simply equivalent if they have the same truth value for all logical

possibilities and it is denote by S₁ (p, q, r, ......) S₂ (p, q, r, ........).

S₁ and S₂ are logically equivalent if they have identical truth tables (means entries in the last column of the truth tables are the same).