Compound Statements in Mathematics

Introduction

A compound statement is a collection of two or more statements joined together using terms like "or," "and," "if-then," and "only if." A statement is the fundamental building block of mathematical reasoning, which is a deductive process. The statements used in reasoning may be compound, meaning they may combine two or more separate statements. In mathematical reasoning issues, some particular words or phrases, such as And, Or, etc., are employed to form compound propositions. These are referred to as connectives.

To answer questions relating to mathematical reasoning, conjunctions like AND, disjunctions like OR, and so on are employed to create compound statements. Connectives are the name given to these words. In this tutorial, we will discuss compound statements.

Mathematical Statements

• We must first recall the fundamentals of mathematics in order to comprehend any mathematical statement. Any mathematical problem we solve has only one possible solution: right or wrong.

• There is no middle ground to the issues. Any mathematical statement would be in a similar situation. A statement in mathematics can only be either true or untrue (false).

• A statement cannot be mathematical if it is projected to be both. We'll use the following three to grasp this better:

India's first prime minister was a female.

The largest animal on Earth is the blue whale.

More intelligent than boys are girls

• While the first assertion is untrue and the second is, the third statement is true for some people and untrue for others. Not all girls are smarter than boys.

• A mathematical assertion is one that can be either true or false. A mathematical assertion is one that can be proven to be true or untrue mathematically and is therefore referred to as such.

Types

Types of Compound Statement:

Based on the connectives utilized throughout the compound statements, the compound statements are categorized. Disjunction statements, conjunction statements, conditional statements, and biconditional statements can all be formed using the connectives "or," "and," "if-then," and "if and only if."

Negation of an Assertion:

• The word no or not is used in the negation. The negation of a statement p is ~𝑝. Denial of a particular statement is known as a negation of the statement. In large part, the negation of a given assertion is regarded as a compound statement.

• Let's look at a straightforward example of negating a statement −

P − Delhi serves as India's capital.

~𝑃 − Delhi is not India's capital.

Disjunctive Assertion

• When two simple statements are combined to make a compound statement that is disjunctive, the connective utilized is "OR."

• A disjunction statement requires that at least one of the claims be true in order for them to be true.

• The OR connective can be used to connect the two straightforward statements P and Q, which is written as 𝑃 𝑉 𝑄. Here, for the compound statement to be true, one of the two statements must be true.

Conjunction Statement:

• The conjunction statement joins two basic statements together by using the conjunction connective, "AND." Both of the statements in the compound statement must be true for it to be true.

• 'And' connectives can be used to join the two simple statements P and Q, and 𝑃 ^ 𝑄 can be used to express the compound statement.

• For a conjunction compound statement to be true, both of the claims in the compound statement must be true.

Conditional Assertion:

• If-then is the connective used for a conditional sentence. Reema will be advanced to the next class if she performs well on the exam.

• Here, the first assertion P may be viewed as the premise, and the second statement, Q can be viewed as the result; the conditional statements for these two straightforward statements P and Q can be written as If P then Q.

• If the hypothesis is correct but the conclusion is incorrect, the conditional compound statement is invalid. However, the conditional assertion is true in all other circumstances.

Bi-Conditional Assertion:

• The connective "If and only if" is used in the biconditional statement and is denoted by the symbol.

• The two statements P and Q are combined to form the compound statement $\mathrm{P\Longleftrightarrow\:Q}$, in which case P is referred to as the antecedent and Q as the consequent.

• The biconditional compound statement, in this case is true if both of the propositions are true or untrue, respectively.

Compound Statement

Several statements are combined to form a compound statement. The terms "and," "or," "if-then," and "if and only if" are used to join the statements to create a compound statement. Connectives are the words that link each of the separate statements together to make a composite statement. Component statements are the individual statements that make up a composite statement.

Solved Examples

1) First assertion: Even numbers can be divided by two. Second assertion: The number two is an even number.

Combining these two propositions results in:

Compound Statement − Even numbers can be divided by two, therefore two is a multiple of two

2) Examples of compound statements.

The sky is blue, and the grass is green.

Either it's cold or sunny

A person is helpful if they are kind

Only when it is divisible by 2, is the number 12 considered to be an even number.

Conclusion

• A compound statement is a collection of two or more statements joined together using terms like "or," "and," "if-then," and "only if." Each component statement of a compound statement can be determined to be true or false with clarity.

• Connectives are the words that link each of the separate statements together to make a composite statement. Component statements are the individual statements that make up a composite statement.

FAQs

1. What do you mean by the compound statement?

A compound statement is one that is created by connecting two simple assertions. Connectives are words that join two simple sentences together, such as "or," "and," "if- then," and "only if."

2. What do you mean by simple statements?

Simple statements are those that are obvious, direct, and absent of modifiers. These problems are easier to solve and don't involve any deductive reasoning.

3. What are disjunction statements?

When two propositions are connected by the connector OR, a disjunction is created.

4. What do you mean by conditional statements?

The symbol "$\mathrm{\Longrightarrow}$" denotes this compound sentence, which employs the connective "if-then."

5. What do you mean by biconditional statements?

The symbol "$\mathrm{\Longleftrightarrow}$" denotes this compound sentence, which uses the connective "if and only if."