. . Compound Statements, Logical Connectives, and Truth Tables Definition . : (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 ( ), ( ), ( ) given in example . are simple statements.
Example for Compond statements Consider the statement, “ 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 : 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 . : (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) . Discrete Mathematics Definition . A statement formula is an expression involving one or more statements connected by some logical connectives. Definition .
: (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. (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 , read as ‘ p conjunction q ’ or ‘ p hat q ’. The truth value of p 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 , read as ‘ p disjunction q ’ or ‘ p cup q ’.The truth value of p is F , whenever both p and q are F and it is T otherwise.