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# Mathematical Logic: Compound Statements, Logical Connectives, and Truth Tables

Any sentence which cannot be split further into two or more statements is called an atomic statement or a simple statement.

Mathematical Logic

Compound Statements, Logical Connectives, and Truth Tables

### Definition 12.8: (Simple and Compound Statements)

Any sentence which cannot be split further into two or more statements is called an atomic statement or a simple statement. If a statement is the combination of two or more simple statements, then it is called a compound statement or a molecular statement. Hence it is clear that any statement can be either a simple statement or a compound statement.

### Example for simple statements

The sentences (1), (2), (3) given in example 12.11 are simple statements.

### Example for Compond statements

Consider the statement, “1 is not a prime number and Ooty is in Kerala”.

Note that the above statement is actually a combination of the following two simple statements:

p : 1 is not a prime number.

q : Ooty is in Kerala.

Hence the given statement is not a simple statement. It is a compound statement.

From the above discussions, it follows that any simple statement takes the value either T or F . So it can be treated as a variable and this variable is known as statement variable or propositional variable. The propositional variables are usually denoted by p, q, r , ....

### Definition 12.9 : (Logical Connectives)

To connect two or more simple sentences, we use the words or a group of words such as “and”, “or”, “if-then”, “if and only if”, and “not”. These connecting words are known as logical connectives.

In order to construct a compound statement from simple statements, some connectives are used. Some basic logical connectives are negation (not), conjunction (and) and disjunction(or).

### Definition 12.10

statement formula is an expression involving one or more statements connected by some logical connectives.

### Definition 12.11: (Truth Table)

A table showing the relationship between truth values of simple statements and the truth values of compound statements formed by using these simple statements is called truth table.

### Definition12.12

(i) Let p be a simple statement. Then the negation of p is a statement whose truth value is opposite to that of p . It is denoted by ¬p, read as not p .The truth value of ¬p is T , if p is F , otherwise it is F .(ii) Let p and q be any two simple statements. The conjunction of p and q is obtained by connecting p and q by the word and. It is denoted by p q , read as ‘ p conjunction q ’ or ‘ p hat q ’. The truth value of p q is T , whenever both p and q are T and it is F otherwise.

(iii) The disjunction of any two simple statements p and q is the compound statement obtained by connecting p and q by the word ‘or’. It is denoted by p q , read as ‘ p disjunction q ’ or ‘ p cup q ’.The truth value of p q is F , whenever both p and q are F and it is T otherwise.

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