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Compound Statements

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Compound Statements - Lesson Summary

A compound statement is a combination of two or more simple statements using words like "and", "or", and so on. Such words are called connecting words.


Conjunction

Consider the statement, "24 is divisible by two and four".

The component statements are:

p: 24 is divisible by 2.

q: 24 is divisible by 4.

A compound statement with the connective word "and" is called a conjunction.

The truth value of a conjunction statement depends on the truth values of its component statements.

The truth value of the conjunction is true, if the truth values of all the component statements are true.

The conjunction is false, if any one of the component statements is false.

Consider the component statements

p: 24 is divisible by 2.

q: 24 is divisible by 4.

Both the component statements are true. Therefore, the compound statement '24 is divisible by 2 and 4' is also true.

Consider the compound statement,

9 is an odd number and a prime number.

The component statements of the compound statement are:

p: 9 is an odd number.

q: 9 is a prime number.

Statement 'p' is true, while statement 'q' is false. Therefore, the compound statement is false.

Every statement with "and" cannot be considered a conjunction.

For example, Jammu and Kashmir is a state of India. In this statement, the word "and" is used to combine two words, and not two statements. Therefore, the given sentence is a statement, but not a conjunction.

Disjunction

A compound statement with the connective "or" is called a disjunction.

A compound statement with "or" is true, when at least one component statement is true. A compound statement with "or" is false, when all its component statements are false.

Ex:

1. Compound statement: 24 is a multiple of 2 and 4.

The component statements are:

p: 24 is a multiple of 2.

q: 24 is a multiple of 4.

Both the component statements are true. Therefore, the compound statement is also true.

2. The component statements of "A triangle has 3 sides or 4 sides," are:

p: A triangle has 3 sides.

q: A triangle has 4 sides.

Statement p is true, while statement q is false. Therefore, the compound statement is true.

3. Compound statement: Plants do not take CO2 or do not give O2.

The component statements are:

p: Plants do not take CO2.

q: Plants do not give O2.

Both the component statements are false. Therefore, the compound statement is also false.

The connective "or" in a compound statement can be either exclusive or inclusive.

Quantifiers

The phrases like "for all", "for every", "for some", "there exists" convey the idea of quantity and refer to a specific collection of objects. Such phrases are called quantifiers.

The quantifiers "for all", "for every" and "for no", are called universal quantifiers.

The universal quantifier is denoted by ".

The quantifiers "there exists", "for some" are called existential quantifiers.
The existential quantifier is denoted by $.

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