Statements: We are familiar with sentences in languages like English. They are sentences that express a certain fact, opinion or idea. They form the basics of communication, written or in speech. As a rule, statements do not ask questions, clarify, or exclaim.

Statements are declarations that set forth particulars or assert. The idea of statements is the same in all languages. Did you know that statements exist beyond languages too? Yes, there are statements in Mathematics. In this article, let us learn about mathematical statements, how to write them, and their types.

What is a Statement?

A statement is a sentence that is either true or false, but not both, simultaneously. Consider:

• Iron man is an Avenger.
• Iron man is not an Avenger.
• Iron man is the best Avenger.
• Is Iron man an Avenger?

Here, \(1\) is true, and \(2\) is false. Some people may think \(3\) is true, while others may argue it is not. Hence, \(3\) is not a statement. Similarly, \(4\) cannot be a statement because it is a question. The same can be applied to mathematical statements too.
Mathematical statements are also called propositions. The principal is to avoid ambiguity.

Why do We Need Statements?

We, humans, are considered to be ’superior’ to other living organisms owing to our reasoning abilities. Statements are the basic unit of mathematical reasoning. When a statement is mentioned, it is considered to be mathematically acceptable. Consider the use of statements as a mathematical communication.
Example:
1. \(30 + y = 73\)
2. \(30 + 43 = 73\)
3. What is the sum of \(30\) and \(43\)?
4. \(73 – 40 = 30\)
Which of these do you think are statements?
Here, \(1\) involves a variable. This means that although the equation is true for one value of \(x,\) it is also false for other values. Hence, \(1\) is not a statement. \(2\) is a statement because it is true. While \(3\) is not a statement because it is a question, \(4\) is a statement because it is false.

Characteristics of a Statement

All statements are sentences. But not all sentences are statements. A sentence is not considered to be a statement if one of the following holds.

• It is an order
• It is a request
• It is a question
• It is an exclamation
• It has pronouns such as he, she, and they
• It has an adverb of time, such as today, yesterday, and tomorrow
• It has an adverb of place, such as here, there, and everywhere

How to Prove a Statement?

While a statement is usually established true using mathematical proofs, it is established false using non-examples or counterexamples. As mathematicians, the first step is to make a conjecture about the statement in consideration through exploration. The goal of exploration is not to establish if the given statement is true or false but simply to investigate the likeliness. A few methods of exploration are listed below.

• Conjecture: Basically, it means to take a guess.
• Apply knowledge: Start from the already acquired/established knowledge.
• Examples: Collect evidence to prove.
• Brainstorming: Comparing notes, articulating, questioning each other and working in groups.

What Are the Types of Statements?

Statements are broadly classified into two types:

• Atomic statements
• Molecular statements

Atomic Statements

Those statements that cannot be broken down into smaller statements are called atomic statements. These are known as simple statements.
Example:
1. \(173\) is an odd number
2. \(2 + 3 = 5\)
3. The square root of \(6\) is \(2\)

Molecular Statements

Statements that are made up of two or more smaller statements are said to be molecular. These are also called compound statements. Here, two atomic statements are combined using a connector. The statements that make a compound statement are called component statements.
Example:
1. \(173\) is an odd and prime number
2. December will be cold or warm
3. The number \(300\) is divisible by \(6\) if and only if the number is divisible by \(3\) and \(2\)

New Statements from Old

The words such as ‘and’, ‘or’, and ‘if and only if’ used in the above compound statements are called connectives. These are used to make new statements using the already existing ones. Logical connectives are mathematically represented using symbols as shown below.

Let’s say \(p\) and \(q\) are two statements. Then:
1. and: \(p \wedge q\)
2. or: \(p \vee q\)
3. negative: \( \sim p\)
4. if and only if: \(p \Leftrightarrow q\)

Conjunction

When two statements are joined using the connective ‘and’, it is called a conjunction. It is represented using the symbol \( \wedge \) and read as “and”.

A conjunction is true only if both statements are true. Even if one of the statements (or both) is false, the conjunction is also false.

Here is the truth table for conjunction.

\(p\)	\(q\)	\(p \wedge q\)
\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)
\(F\)	\(T\)	\(F\)
\(F\)	\(F\)	\(F\)

Example of Conjunction

\(p:\) Some integers are negative numbers.
\(q:\) All squares are rectangles.
Then, \(p \wedge q:\) Some integers are negative numbers, and all squares are rectangles.
Here, we know that \(p\) and \(q\) are true. Therefore, the conjunction \(p \wedge q\) is also true.

Disjunction

A disjunction is a compound statement that has two simple statements connected by the logical operator ‘or’. It is represented using the symbol \( \vee \) and read as “or”.
At least one of the statements in a disjunction must be true for the disjunction to be true. It is false if none of the statements are true.
Truth table for disjunction is as shown below.

\(p\)	\(q\)	\(p \vee q\)
\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(T\)
\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(F\)

Example of Disjunction

\(p:\) Some integers are negative numbers.
\(q:\) All rectangles are squares.
Then, \(p \vee q:\) Some integers are negative numbers, or all rectangles are squares.
Here, \(p\) is true, and \(q\) is false. Hence, according to the truth table, the disjunction \(p \vee q\) is also true.

Negation

In Mathematics, the term negation means the opposite. For a statement p, the negation is written as \( \sim p,\) and read as ‘not \(p\)’.
For this logical connector, if \(p\) is positive, then \( \sim p\) is negative, and vice versa.
Let us now show this in a truth table.

\(p\)	\( \sim p\)
\(T\)	\(F\)
\(F\)	\(T\)

Example:
\(p:\) Sun is a star.
Then,\( \sim p:\) Sun is not a star.

Negation of a Conjunction

We know that conjunction is represented as \(p \wedge q.\) The negation of a conjunction is represented as,
\( \sim \left( {p \wedge q} \right) = \sim p \vee \sim q\)
Thus, we can say that the negation of a conjunction is the disjunction of the negation of the individual statements \(p\) and \(q.\)

Negation of a Disjunction

We know that a disjunction is represented as \(p \vee q.\) The negation of a disjunction is represented as,
\( \sim \left( {p \vee q} \right) = \sim p \wedge \sim q\)
Thus, we can say that the negation of a disjunction \({p \vee q}\) is the conjunction of the negation of the individual statements \(p\) and \(q.\)

Negation of a Negation

The negation of a statement \(p\) is represented as \( \sim p.\) The negation of negation can be represented as,
\( \sim \left( { \sim p} \right) = p\)
Hence, we can say that the negation of negation of a statement results in the statement itself.

Conditional Statement

This is the commonly used type of equation in Mathematics. The statements of the form, “If \(p,\) then \(q\)” is called a conditional statement. Here, the truth value of the conditional statement depends on the truth value of the individual statements. The ‘if’ condition is called the hypothesis, and ‘then’ is called the conclusion. Conditional statements are represented using the symbol \( \to .\)
The truth table of the conditional statement is as follows.

\(p\)	\(q\)	\(p \to q\)
\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)
\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)

Observe that the conditional statement \(p \to q\) has its own truth value. The truth value of a conditional statement is false only when the hypothesis is true, and the conclusion is false.
Conditional statement \(p \to q\) can be expressed in different ways. Some of them are listed below.
1. If \(p,\) then \(q\)
2. \(q\) if \(p\)
3. \(p\) only if \(q\)
4. \(p\) is sufficient for \(q\)
5. \(q\) is necessary for \(p\)

Example of Conditional Statement

\(p:x\) is a positive real number
\(q:{x^2} + 8x\) is a real number
Then, \(p \to q:\) If \(x\) is a positive real number then, \({x^2} + 8x\) is a real number

Biconditional Statement

Consider the example,
The number \(300\) is divisible by \(6\) if and only if the number is divisible by \(3\) and \(2\)
Here,
\(p:\) The number \(300\) is divisible by \(6\)
\(q:\) The number is divisible by \(3\) and \(2\)
When two statements in a compound statement have the connective ‘if and only if’, it is called a biconditional statement. It is represented as \(p \leftrightarrow q\) in symbolic form.

Solved Examples

Q.1. Check if the given sentences are statements. Give reasons.
a. Every set is a finite set.
b. Mathematics is an easy subject for grade \(11.\)
c. \(18\) is more than \(16.\)
d. How far is Delhi from Chennai?
Ans: a. Every set is a finite set.
The truth value of this sentence is false. Hence, this is a statement.
b. Mathematics is an easy subject for grade \(11.\)
There is no specific truth value for this sentence. Hence, this is not a statement.
c. \(18\) is more than \(16.\)
The truth value of this sentence is true. Hence, this is a statement.
d. How far is Delhi from Chennai?
This is not a statement because this is a question.

Q.2. Write the negation of the given statements.
a. Every whole number is less than \(0.\)
b. \(p:\) For every real number \(x,{x^2} > 2.\)
c. \(q:\) All students study Mathematics in elementary.
Ans: a. Every whole number is more than \(0.\)
b. \(p:\) For every real number \(x,{x^2} < 2.\)
c. \(q:\) All students do not study Mathematics in elementary.

Q.3.Find the component statements of the given compound statements and their connectives.
a. Number \(3\) is prime, or it is odd.
b. All integers are positive or negative.
c. \(100\) is divisible by \(3,11,\) and \(5.\)
Ans: a. Number \(3\) is prime, or it is odd.
\(p:\) Number \(3\) is prime.
\(q:\) Number \(3\) is odd.
Connective: or
b. All integers are positive or negative.
\(p:\) All integers are positive.
\(q:\) All integers are negative.
Connective: or
c. \(100\) is divisible by \(3,11,\) and \(5.\)
\(p:\) \(100\) is divisible by \(3\)
\(q:\) \(100\) is divisible by \(11\)
\(r:\) \(100\) is divisible by \(5.\)
Connective: and

Q.4. Translate the statement into a symbolic form: \(2\) and \(3\) are prime numbers
Ans:
\(p:2\) is a prime number.
\(q:3\) is a prime number.
Therefore, \(2\) and \(3\) are prime numbers that can be represented as \(p \wedge q,\) where the connective is ‘and’.

Q.5. Check whether the following statement is true: “if \(a\) and \(b\) are odd integers, then \(ab\) is an odd integer”.
Ans:
We know that the product of two or more odd numbers is always odd.
Therefore, if \(a\) and \(b\) are odd integers, then \(ab\) is an odd integer. Hence, the statement is true.

Q.6.Given:
a. \(p:12\) is prime
b. \(q:17\) is prime
c. \(r:19\) is composite
Write a sentence for each disjunction given below. Write the truth value.
1. \(p \vee q\)
2. \(p \vee r\)
3. \(q \vee r\)
Ans:
From the given statements, we can write:

Statement	Truth Value
\(p:12\) is prime	False
\(q:17\) is prime	True
\(r:19\) is composite	False

Therefore, we have

Disjunction	Statement	Truth Value
\(p \vee q\)	\(12\) or \(17\) is prime	True
\(p \vee r\)	\(12\) is prime, or \(19\) is composite	False
\(q \vee r\)	\(17\) is prime, or \(19\) is composite	True

Summary

Statements are sentences that are either true or false, but not both. They are of two types: simple or atomic and compound or molecular. While simple statements cannot be broken, compound statements can be broken into two or more simple statements. A compound statement consists of two or more simple statements joined by a connective. Statements with ‘and’ operator are called conjunctions, and that with ‘or’ operator are called disjunctions. Each of them has a truth value that depend on the truth value of the component statements. While a conditional statement is of the form: ‘If \(p,\) then \(q’,\) a biconditional statement is \(‘p\) if and only if \(q’.\)

Frequently Asked Questions (FAQs)

Q.1. What do you mean by a statement?
Ans: Any sentence that is either true or false, but not both, is called a statement.
Example:
\(p:3\) is a factor of \(297.\)
\(q:\) All integers are positive or negative.

Q.2. What is a simple statement example?
Ans: Statements that cannot be broken down are called simple statements.
Example:
1. Delhi is the capital of India.
2. India is a peninsula.
3. The sum of \(93\) and \(42\) is \(125.\)

Q.3. What are biconditional statements?
Ans: Compound statements that use the connective ‘if and only if’ are called biconditional statements.
Example:
1. A quadrilateral is a square if and only if it has congruent sides and angles.
2. A triangle is equilateral if and only if it is equiangular.

Q.4. What is the negation of the statement: “If I become a teacher, then I will open a school”?
Ans: The negation is: “If I do not become a teacher, then I will not open a school”.

Q.5. How do you write the negation of a statement?
Ans: Negation statements convey the opposite of the given statement. It is represented using the \( \sim \) symbol.
Example:
\(p:21 > 17\)
\( \sim p:21 < 17\)

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