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

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.