Compound statements and Truth table

Truth Tables

A table is indicating the truth values of one or more statements is called a truth table.

Example:

The table represents the case when p is true and q is false.

Logical connectives and compound statements

Normally propositions are joined by 'OR', 'AND', 'NOT', 'IF .. THEN..' and 'IF AND ONLY IF' called the logical connectives. They are symbolised as follows:

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.

Following are some examples of compound statements:

• 2+3=7 or 6 < 9

• If 2+3=7, then Sun rises in the west.

The symbolic representation of compound statements are illustrated below:

Let

p : A is intelligent.

q : B passes the exam.

Conjunctions

If p and q are two statements, then the compound statement “p and q", is called the conjunction denoted by "p Ùq". pÙq is true when both p and q are true.

Disjunctions

If p and q are two statements, then the compound statement “p or q” is called the disjunction, denoted by "p v q", p v q is false when both p and q are false.

Conditional statement

If p and q are two statements, then the compound statement “If p then q" is called the compound statement denoted by "p Þ q", p Þ q is false when p is true and q is false.

Note:

Biconditional statement

If p and q are two statements then the compound statement “p if and only if q", is called the biconditional statement denoted by "p Û q". "p Û q" is true when both p and q are true or when both p and q are false.

Negation

If p is any statement, then the statement “not p” is called the negation of p denoted by "~p".

~p is true, when p is false and ~p is false, when p is true.

Note:

Suppose a compound proposition is given, we first split it into simple propositions containing a single connective. Using the rules discussed above, we construct the truth table in the form of columns and the last column gives the truth value of the given proposition for different combinations of the truth values of its components.

Tautology

A compound statement is said to be a tautology, if it is always true for all possible combinations of the truth values of its components.

A tautology is also called a theorem or a logically valid statement pattern.

Contradiction

A compound statement is said to be a contradiction, if it is always false for all possible combinations of the truth values of its components.

Note:

(i) The negation of a tautology is a contradiction.

(ii) The negation of a contradiction is a tautology.