Compound Statements

Compound statement is a group of two or more statements connected using words such as 'or', 'and', 'if then', 'if and only if'. Each statement of a compound statement is a component statement, which can be clearly decided as a true or false statement. The individual statements are represented as p, q and the compound statements are represented as p v q, p ^ q, p ⇒ q, p ⇔ q.

Let us learn more about the compound statement, types of compound statements, their truth tables, with the help of examples, FAQs.

1.	What Are Compound Statements?
2.	Types of Compound Statements
3.	Truth Table of Compound Statements
4.	Examples on Compound Statements
5.	Practice Questions
6.	FAQs on Compound Statements

What Are Compound Statements?

Compound statement is made up of two or more statements. The statements are combined using words such as 'and', 'or', 'if then', 'if and only if' to form a compound statement. These words used to connect each of the individual statements to form a compound statement are called connectives. Each statement of the compound statement is called a component statement.

Compound Statements

Examples of Compound Statements:

The grass is green and the sky is blue.
It is cold or it is sunny.
If a person is kind then he is helpful.
The number 12 is an even number if and only if it is divisible by 2.

Compound statements are generally formed from simple statements which are represented as p, q, and the compound statements are represented as p v q, p ^ q, p ⇒ q, p ⇔ q. The symbols used to connect the statements p, q are v, ^, ⇒, ⇔ represent the words 'or', 'and', 'if then', 'if and only if', and are referred to as connectives. The words 'or', 'and' are useful to form a compound statement, but every statement having these words 'or', 'and' need not be a compound statement.

Types Of Compound Statements

The compound statements are classified based on the connectives used across the compound statements. The connectives of 'or', 'and', 'if then', 'if and only if', are used to form disjunction statements, conjunction statements, conditional statements, and biconditional statements. Let us check in detail about each of these compound statements.

Negation of a Statement: The negation uses the word no, not. For a statement p, its negation is ~p. The negation of a given statement is the denial of a given statement. The negation of a given statement is to a good extent considered as a compound statement. Let us take a simple example of negation of a statement: P: Delhi is the capital of India. ~P: Delhi is not the capital of India.
Disjunction Statement: The connective used for two simple statements to form a compound statement which is disjunction is 'OR'. In a disjunction statement, any one of the statements must be true for the disjunction statement to be true. The two simple statements represented as P and Q can be connected using OR connective and is written as P V Q. Here any of the two statements should be true for the compound statement to be true.
Conjunction Statement: The compound statement of conjunction uses the connective 'AND' for connecting two simple statements. For this compound statement both the statements must be true for the compound statement to be true. The two simple statements P and Q can be connected using 'And' connective and the compound statement can be written as P ^ Q. For a conjunction compound statement, both the statements should be true for the compound statement to be true.
Conditional Statement. The connective used for a conditional statement is if then. If Reema does well in the test then She will be promoted to the next class. Here the first statement P can be taken as the hypothesis, and the second statement Q can be taken as the conclusion, we can write condition statements of these two simple statements P, Q as If P then Q. The conditional compound statement does not hold true if the hypothesis is true and the conclusion is false. But in all other situations, the conditional statement is true.
Bi Conditional Statement: The biconditional statement uses the connective 'If and only if'., which is represented by the symbol ⇔. The two statements P and Q are represented as a compound statement P ⇔ Q, and here the first statement P is called the antecedent and the second statement Q is called the consequent. Here the biconditional compound statement is true if both the statements are either true or both are either false.

Truth Tables of Compound Statement

The truth value of a compound statement depends on the truth value of the individual statements and also on the connective used to form the compound statement. The truth tables of the different types of compound statements are as follows.

Disjunction Truth Table uses the connective 'or' to form the compound statement. Here even if one of the individual statements is true, then the compound statement also holds true.

P	Q	P V Q
T	F	T
T	T	T
F	T	T
F	F	F

Conjunction Truth Table uses the connective 'and' to form the compound statement. Here the compound statement is true only if both the individual statements are true. Even if one of the individual statements is false, then the compound statement is considered as a false statement.

P	Q	P ^ Q
T	F	F
T	T	T
F	T	F
F	F	F

Conditional Truth Table uses If-then connective, which is represented as ⇒. Here the statement p is referred to as a hypothesis and the statement q is referred to as conclusion, and the compound statement is true if the conclusion is true, irrespective of the hypothesis. Also, the compound statement is true if both the hypothesis and the conclusion are false.

P	Q	P ⇒ Q
T	F	F
T	T	T
F	T	T
F	F	T

IBiconditional Truth Table used the connective 'if and only if' and is represented as ⇔. Here the first statement p is referred to as antecedent and the second statement q is referred to as consequent. The biconditional compound statement is true if the second statement, the consequent is false.

P	Q	P ⇔ Q
T	F	T
T	T	F
F	T	F
F	F	T

☛Related Topics

Mathematical Reasoning
Principles of Mathematical Induction
Binomial Theorem
Binomial Expansion Formulas

Examples on Compound Statements

Example 1: For the compound statement: If it is raining then it will be very cold", write the converse, inverse, and contrapositive statements. Solution: Conditional Statement: P → Q : If it is raining then it will be very cold. Converse Statement: Q → P: If it is very cold then it will be raining. Inverse Statement: ~P → ~Q: If it is not raining then it will not be very cold. Contrapositive Statement:~Q → ~P: If it is not very cold then it is not raining.

Example 2: What is the compound statement which can be formed from the statements P: you go regularly to school. and q: you get good marks. ? Solution: The two given statements are: P: You go regularly to school. Q: you get good marks. The four possible connectives which can be used here are and, or, if then, if and only if. Let us form the four compound statements. Conjunction Statement: (And connective) You go regularly to school and you get good marks. Disjunction Statement: (Or Connective) You go regularly to school or you get good marks. Conditional Statement: (If then connective) If you go regularly to school then you get good marks. Biconditional Statement: (If and only if connective) You go regularly to school if and only if you get good marks. Among the four statements, the conditional statement works well as the second statement is dependent on the first statement.

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