Mathematics · Class 11 · Mathematical Reasoning · Term 3

Statements (Propositions)

Learn to differentiate between sentences that are commands, questions, or exclamations, and those that are mathematically acceptable statements (propositions) which can be judged as true or false.

TL;DR:Let's begin our journey into mathematical reasoning by learning the special language mathematicians use. We will discover what makes a simple sentence powerful enough to be called a 'statement'.

• CBSE Learning Outcomes
• NCERT Class 11: Chapter 14 - Mathematical Reasoning

About This Topic

This topic, 'Statements', serves as the foundational entry point into the chapter on Mathematical Reasoning in the Class 11 curriculum, as prescribed by NCERT. It marks a significant shift for students, moving them from the computational and algebraic manipulation they are accustomed to, towards the abstract and rigorous world of mathematical logic. The primary goal is to teach students to differentiate between the varied sentences of everyday language and the precise, unambiguous sentences that are acceptable in a mathematical context. These acceptable sentences are called 'statements' or 'propositions'.

The core idea is that a mathematical statement must be a declarative sentence which can be definitively judged as either true or false, but not both. This concept is the bedrock upon which all logical deduction, proof techniques, and higher mathematics are built. This topic lays the groundwork for understanding logical connectives (like 'and', 'or', 'not'), implications ('if-then'), contrapositives, and quantifiers ('for all', 'there exists'), which are crucial for constructing and deconstructing mathematical proofs. By mastering this initial step, students develop a critical eye for precision and an appreciation for the formal language of mathematics.

Key Questions

1. Explain the criteria that a sentence must meet to be considered a mathematical statement.
2. Analyse the sentence 'Mathematics is fun' and justify why it is not a valid mathematical statement.
3. Compare the sentences 'The sum of two even numbers is even' and 'Close the door' in terms of logical reasoning.

Learning Objectives

• Define a mathematical statement and its truth value.
• Distinguish between sentences that are statements and those that are not (questions, commands, etc.).
• Analyse a given sentence to determine if it is a mathematically acceptable statement.
• Identify the truth value of simple, well-defined mathematical statements.
• Explain why ambiguity or subjectivity disqualifies a sentence from being a statement.

Key Vocabulary

Statement (or Proposition)	A declarative sentence which is either true or false, but not both.
Truth Value	The property of a statement being either true (T) or false (F).
Declarative Sentence	A sentence that makes a declaration or asserts a fact.
Open Sentence	A sentence containing one or more variables, which becomes a statement when the variables are replaced by specific values.
Ambiguous Sentence	A sentence whose meaning or truth value is not clear or can be interpreted in more than one way.

Watch Out for These Misconceptions

Common Misconception

If a sentence is false, it cannot be a mathematical statement.

What to Teach Instead

A mathematical statement is any declarative sentence that has a definite truth value. This value can be either 'true' OR 'false'. For example, 'The Earth is flat' is a perfectly valid statement; its truth value is false.

Common Misconception

Any sentence containing numbers or variables is a mathematical statement.

What to Teach Instead

A sentence must be declarative. 'Is 5 greater than 2?' is a question, not a statement. Also, a sentence with a variable like 'x + 2 = 5' is an 'open sentence', not a statement, because its truth depends on the value of x.

Common Misconception

Opinions or subjective sentences can be statements if I believe them to be true.

What to Teach Instead

A statement's truth value must be objective and not depend on personal opinion. 'Mathematics is a difficult subject' is not a statement because its truth varies from person to person.

Active Learning Ideas

Think-Pair-Share

Statement or Not? Card Sort

Prepare cards with various sentences (commands, questions, opinions, facts, paradoxes). In small groups, students sort these cards into three piles: 'Is a Statement', 'Not a Statement', and 'Unsure/Debatable'.

20 min·Small Groups

Think-Pair-Share

Truth Value Hunt

Provide a list of valid mathematical statements. In pairs, students must determine the truth value (True or False) of each statement, providing a brief justification for their answer.

15 min·Pairs

Think-Pair-Share

Write and Swap

Each student writes three sentences: one that is a true statement, one that is a false statement, and one that is not a statement. They then swap with a partner who must correctly identify each one.

10 min·Individual

Real-World Connections

• In computer science, the logic of programming relies on statements that evaluate to TRUE or FALSE to control the flow of a program (e.g., in 'if-else' conditions).
• In legal systems, lawyers construct arguments by linking a series of statements (evidence, facts) to reach a logical conclusion of 'guilty' or 'not guilty'.
• In digital circuit design, logic gates (AND, OR, NOT) process inputs (0s and 1s, representing false and true) based on the principles of mathematical logic.
• In journalism, fact-checking involves verifying whether statements made in a report are true or false before publication.
• The scientific method is based on forming a hypothesis, which is a testable statement, and then conducting experiments to determine its truth value.

Assessment Ideas

Exit Ticket

Use an exit ticket: provide three sentences (e.g., a question, an opinion, and a statement) and ask students to identify the statement and briefly explain why the other two are not.

Discussion Prompt

During class discussion, use targeted questioning. Ask 'Why is this not a statement?' or 'How could we change this question into a statement?' to check for understanding.

Quick Check

A short quiz containing a mix of sentences where students must identify which are statements. For those that are statements, they must also assign a truth value (True/False/Cannot be determined).

Frequently Asked Questions

What is the main difference between a regular English sentence and a mathematical statement?

While all mathematical statements are sentences, not all sentences are statements. A mathematical statement must be a declarative sentence that can be objectively and unambiguously classified as either true or false. Questions, commands, exclamations, and opinions are sentences but not statements.

Why is 'The weather is pleasant today' not a statement?

This sentence is not a statement for two reasons. First, 'pleasant' is subjective and depends on personal opinion. Second, 'today' is ambiguous; the truth of the sentence depends on which day it is said. A mathematical statement must have a consistent truth value.

Is 'x is a real number' a statement?

No, this is an open sentence. It is not a statement because we do not know the value of 'x'. If we replace 'x' with a specific value, it becomes a statement. For example, '5 is a real number' is a true statement, and '√-1 is a real number' is a false statement.