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| Propositions: Definition of a Mathematical Statement | Identifying Mathematical Statements | Truth Values of Statements (True or False) |
Statements and Propositions: Fundamentals
Propositions: Definition of a Mathematical Statement
Mathematical reasoning, also known as logic in mathematics, provides a structured way to deduce conclusions from given assumptions. At the heart of this structure lies the concept of a mathematical statement or proposition. These are the fundamental building blocks upon which all logical arguments and proofs are constructed.
Definition of a Statement (Proposition)
A mathematical statement (or proposition) is formally defined as a declarative sentence that is unequivocally either true (T) or false (F), but critically, it cannot be both true and false simultaneously within the same context.
In simpler terms, when you encounter a sentence, you must be able to determine, without any ambiguity or room for personal opinion, whether that sentence is definitively true or definitively false. If you can do that, it's a mathematical statement.
Essential Characteristics of a Mathematical Statement
Let's break down the key features that a sentence must possess to be considered a mathematical statement:
- Must be a Declarative Sentence: This is the primary linguistic requirement. A declarative sentence makes an assertion or states a fact. It stands in contrast to other types of sentences like questions (interrogative), commands (imperative), or exclamations. Only sentences that declare something can potentially be judged as true or false.
- Must Have a Definite Truth Value: The sentence must be either true or false. This truth value should be objective and verifiable based on established mathematical definitions, axioms, logical rules, or agreed-upon facts (in the context of applying logic to real-world scenarios). Subjective opinions, beliefs, or feelings do not qualify as having a definite truth value in this logical sense.
- The Truth Value Must Be Unambiguous: There should be no uncertainty about whether the statement is true or false. A sentence whose truth value depends on external factors that are not specified (like the value of a variable, or the specific time or place) is usually not a statement on its own. Such sentences are often called "open sentences" or "predicates".
- Cannot Be Both True and False: This reflects the Law of Excluded Middle (a statement is either true or false) and the Law of Non-Contradiction (a statement cannot be both true and false). Sentences that violate this principle, such as paradoxes, are generally excluded from the set of mathematical statements.
These characteristics ensure that mathematical logic deals with clear-cut assertions that can be evaluated precisely, forming a reliable foundation for rigorous proofs and deductions.
Example 1. Identify which of the following sentences are mathematical statements. Provide a brief reason for your conclusion.
(i) Mumbai is the largest city in India.
(ii) $10 \div 2 = 5$.
(iii) What is the square root of 9?
(iv) $y - 3 = 11$.
(v) Every prime number greater than 2 is odd.
(vi) Look at the board.
(vii) $\pi$ is a rational number.
(viii) For all integers $n$, $n^2 \geq 0$.
(ix) This tea is too sweet.
Answer:
(i) "Mumbai is the largest city in India."
This is a declarative sentence. Based on population statistics, Mumbai is indeed the largest city in India. This is an objectively verifiable fact, making the statement True.
Therefore, this is a mathematical statement.
(ii) "$10 \div 2 = 5$."
This is a declarative sentence stating a mathematical equality. In standard arithmetic, $10$ divided by $2$ equals $5$. This is objectively True.
Therefore, this is a mathematical statement.
(iii) "What is the square root of 9?"
This is an interrogative sentence (a question). It asks for information and does not assert anything that can be true or false.
Therefore, this is not a mathematical statement.
(iv) "$y - 3 = 11$."
This is a declarative sentence stating an equality involving a variable $y$. Its truth depends on the value of $y$. If $y=14$, it is True ($14-3=11$). If $y=10$, it is False ($10-3=7 \neq 11$). Since its truth value is not fixed, it is not unambiguously true or false.
Therefore, this is not a mathematical statement. It is an open sentence.
(v) "Every prime number greater than 2 is odd."
This is a declarative sentence asserting a property for all prime numbers greater than 2. Prime numbers greater than 2 are 3, 5, 7, 11, 13, ... All of these are odd numbers. This statement is objectively True.
Therefore, this is a mathematical statement.
(vi) "Look at the board."
This is an imperative sentence (a command). It instructs someone to perform an action and does not assert anything that can be true or false.
Therefore, this is not a mathematical statement.
(vii) "$\pi$ is a rational number."
This is a declarative sentence stating a property of the mathematical constant $\pi$. $\pi$ is defined as the ratio of a circle's circumference to its diameter, and it is a well-established fact in mathematics that $\pi$ is an irrational number (it cannot be expressed as a simple fraction $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$). Thus, the statement "$pi$ is a rational number" is objectively False.
Therefore, this is a mathematical statement (a false one).
(viii) "For all integers $n$, $n^2 \geq 0$."
This is a declarative sentence. It asserts a property for every possible integer $n$. The square of any integer (positive, negative, or zero) is always non-negative. This statement is objectively True.
Therefore, this is a mathematical statement. (Note the presence of the quantifier "For all", which makes the open sentence "$n^2 \geq 0$" into a statement).
(ix) "This tea is too sweet."
This is a declarative sentence. However, the term "too sweet" is subjective and depends entirely on the individual's palate and preference. There is no objective measure to determine if this statement is universally true or false.
Therefore, this is not a mathematical statement.
Summary: Statements vs. Non-Statements
To solidify the concept, let's look at a table summarising different types of sentences and whether they qualify as mathematical statements.
| Sentence | Type | Declarative? | Has Unambiguous Objective Truth Value? | Is it a Mathematical Statement? |
|---|---|---|---|---|
| The sun rises in the East. | Fact/Observation | Yes | Yes (True) | Yes |
| $2 \times 3 = 7$. | Mathematical assertion | Yes | Yes (False) | Yes |
| $x^2 + 1 = 0$. | Open sentence (with variable) | Yes | No (depends on $x$) | No |
| $\sqrt{2}$ is irrational. | Mathematical fact | Yes | Yes (True) | Yes |
| Sing a song. | Command (Imperative) | No | N/A | No |
| What a wonderful day! | Exclamation | No | N/A | No |
| Cricket is the best sport. | Subjective Opinion | Yes | No (subjective) | No |
| For every real number $r$, $r^2 \geq 0$. | Quantified assertion | Yes | Yes (True) | Yes |
Competitive Exam Pointer
In competitive examinations, questions often involve identifying statements. Be careful with sentences containing:
- Variables: Like "$x+5=12$" or "$n$ is a multiple of 3". These are open sentences unless the variable is quantified (e.g., "There exists an $x$ such that $x+5=12$" or "For all integers $n$, $n$ is a multiple of 3").
- Pronouns/Ambiguous references: "He is a doctor." (Who is "he"?)
- Subjective words: "good", "bad", "beautiful", "ugly", "fun", "easy", "difficult".
- Commands, questions, exclamations: These are straightforwardly not statements.
A sentence like "The square of an even number is even" is a statement (and it's True), because it refers to a property of a defined set of numbers, irrespective of a specific variable.
Identifying Mathematical Statements
To identify whether a given sentence is a mathematical statement (proposition), we apply the definition strictly. We must determine if the sentence is a declarative sentence that is definitively and unambiguously either true or false, but not both. The truth or falsity should be verifiable based on established facts, definitions, or logical principles, not on personal opinion or undefined conditions.
Criteria for Identification Revisited
Remember the two crucial criteria:
- Is the sentence a declaration that asserts something?
- Can you determine, without any doubt or subjectivity, if the sentence is True or False?
If both answers are "Yes", it's a statement. If either is "No", it's not.
Let's analyze more examples:
Example 1. Determine whether the following sentences are mathematical statements. Give reasons for your answer.
(i) The square root of 2 is an irrational number.
(ii) Please give me that book.
(iii) All prime numbers are odd.
(iv) $x^2 - 1 = 0$.
(v) Tomorrow is Friday.
(vi) Wow! What a beautiful sunset.
(vii) There is no rain without clouds.
(viii) The sum of interior angles of a triangle is $180^\circ$.
Answer:
(i) "The square root of 2 is an irrational number."
- Declarative? Yes, it asserts a property of the number $\sqrt{2}$.
- Unambiguous Truth Value? Yes. It is a well-known theorem in mathematics that $\sqrt{2}$ cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers ($q \neq 0$). Therefore, $\sqrt{2}$ is irrational. This statement is objectively True.
Result: This is a mathematical statement.
(ii) "Please give me that book."
- Declarative? No. This is a request or a command (an imperative sentence). It does not make an assertion that can be true or false.
Result: This is not a mathematical statement.
(iii) "All prime numbers are odd."
- Declarative? Yes, it makes a claim about all prime numbers.
- Unambiguous Truth Value? Yes. We can verify this statement. The number 2 is a prime number (its only positive divisors are 1 and 2), and it is an even number. Since we found one case (the number 2) for which the statement is false, the entire statement "All prime numbers are odd" is objectively False.
Result: This is a mathematical statement (a false one).
(iv) "$x^2 - 1 = 0$."
- Declarative? Yes, it asserts an equality.
- Unambiguous Truth Value? No. The truth of this sentence depends on the value assigned to the variable $x$. If $x=1$ or $x=-1$, the equation is true. For any other value of $x$, it is false. Without knowing the value of $x$, or without a quantifier (like "For all $x$..." or "There exists an $x$ such that..."), the truth value is not fixed.
Result: This is not a mathematical statement (it's an open sentence or predicate).
(v) "Tomorrow is Friday."
- Declarative? Yes, it makes a prediction.
- Unambiguous Truth Value? No. The truth value of this sentence changes depending on the day it is spoken or written. If "today" is Thursday, it's True. If "today" is Saturday, it's False. Because its truth depends on the specific day, it is ambiguous without a fixed point of reference.
Result: This is not a mathematical statement.
(vi) "Wow! What a beautiful sunset."
- Declarative? No. This is an exclamation expressing emotion. It does not make an assertion that can be judged as true or false.
Result: This is not a mathematical statement.
(vii) "There is no rain without clouds."
- Declarative? Yes, it asserts a relationship between rain and clouds.
- Unambiguous Truth Value? Yes. Based on meteorological understanding, clouds are necessary for rain formation. While there might be nuances depending on specific atmospheric conditions (like virga, where rain evaporates before reaching the ground), the general scientific principle makes this statement objectively True within that context.
Result: This is a mathematical statement.
(viii) "The sum of interior angles of a triangle is $180^\circ$."
- Declarative? Yes, it makes a claim about triangles.
- Unambiguous Truth Value? Yes. This is a fundamental theorem in Euclidean geometry, which is the geometry typically studied in school. This statement is objectively True. (Note: In non-Euclidean geometries, this statement might be false, but within the standard context, it's true).
Result: This is a mathematical statement.
Distinguishing Statements from Non-Statements: Key Takeaways
Identifying mathematical statements boils down to checking for these two core properties. If a sentence lacks either property (it's not declarative, or its truth value is ambiguous/subjective), it's not a statement in the context of logic.
Here's a summary table of sentence types:
| Sentence Type | Example | Is it a Statement? | Reason |
|---|---|---|---|
| Declarative, Unambiguously True | $2+2=4$ | Yes | Declares a universally true fact. |
| Declarative, Unambiguously False | Kolkata is the capital of Pakistan. | Yes | Declares a universally false fact. |
| Declarative, Ambiguous (Open Sentence) | $x$ is an even number. | No | Truth depends on the value of $x$. |
| Declarative, Subjective | Indian food is delicious. | No | Truth depends on personal taste. |
| Interrogative | Are you coming? | No | It's a question, not an assertion. |
| Imperative | Submit the form. | No | It's a command, not an assertion. |
| Exclamatory | What a mess! | No | Expresses emotion, not an assertion. |
Truth Values of Statements (True or False)
As established, a defining characteristic of a mathematical statement (or proposition) is that it must possess a definite truth value. There are precisely two possible truth values:
- True (T): The statement corresponds to fact or is consistent with the established rules or definitions of the system being considered (e.g., arithmetic, geometry, general knowledge if the statement is outside pure math).
- False (F): The statement does not correspond to fact or is inconsistent with the established rules or definitions.
Assigning Truth Values
Assigning the correct truth value to a statement is the very first step in analyzing its role in a logical argument. The truth value is determined by evaluating the assertion made by the statement against reality, definitions, or mathematical principles. It's not about whether we *know* the truth value right now, but whether a definite truth value *exists* objectively.
For example, the statement "There are an infinite number of twin primes" is a statement. As of now, mathematicians haven't proven it true or false, but it is objectively one or the other. Its truth value is definite, even if currently unknown to us.
Example 1. Determine the truth value of the following mathematical statements:
(i) $p$: Delhi is the capital of India.
(ii) $q$: $3 + 7 = 9$.
(iii) $r$: Every rectangle is a square.
(iv) $s$: The number 5 is a prime number.
(v) $t$: $\sqrt{-1}$ is a real number.
(vi) $u$: The sum of two odd numbers is odd.
(vii) $v$: A square has exactly four diagonals.
Answer:
(i) Statement $p$: "Delhi is the capital of India."
This statement asserts a fact about the political geography of India. Based on current knowledge, New Delhi is the capital of India. While sometimes "Delhi" is used loosely to refer to the National Capital Territory, in the context of states and capitals, New Delhi is precise. Assuming the statement refers to the political capital, it is true.
Truth Value of $p$: T
(ii) Statement $q$: "$3 + 7 = 9$."
This statement asserts a mathematical equality. Performing the addition $3 + 7$, we get $10$. The statement claims this sum is $9$, which is incorrect.
Truth Value of $q$: F
(iii) Statement $r$: "Every rectangle is a square."
This statement makes a claim about geometrical shapes. By definition, a rectangle is a quadrilateral with four right angles. A square is a rectangle with all four sides equal. While all squares are rectangles, not all rectangles are squares (e.g., a rectangle with unequal adjacent sides is not a square). The statement claims that *every* rectangle is a square, which is false because we can find counterexamples.
Truth Value of $r$: F
(iv) Statement $s$: "The number 5 is a prime number."
This statement pertains to number theory. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The number 5 is greater than 1, and its only positive divisors are 1 and 5. Therefore, 5 fits the definition of a prime number. The statement is True.
Truth Value of $s$: T
(v) Statement $t$: "$\sqrt{-1}$ is a real number."
This statement relates to number systems. The set of real numbers includes all rational and irrational numbers. Real numbers squared result in a non-negative number. The square root of -1 is denoted by $i$ and is the fundamental unit of imaginary numbers. It is specifically *not* a real number.
Truth Value of $t$: F
(vi) Statement $u$: "The sum of two odd numbers is odd."
Let's test this with examples. Take two odd numbers, say 3 and 5. Their sum is $3+5=8$, which is an even number. Since we found a counterexample, the statement that the sum of *any* two odd numbers is odd is false. Let's try to prove it algebraically: An odd number can be written as $2k+1$ for some integer $k$. The sum of two odd numbers would be $(2k_1 + 1) + (2k_2 + 1) = 2k_1 + 2k_2 + 2 = 2(k_1 + k_2 + 1)$. Since $k_1$ and $k_2$ are integers, $k_1+k_2+1$ is also an integer. The form $2 \times (\text{integer})$ means the sum is always even.
Truth Value of $u$: F
(vii) Statement $v$: "A square has exactly four diagonals."
Let's consider a square. A square has 4 vertices. A diagonal connects two non-adjacent vertices. Let the vertices be A, B, C, D in order. The non-adjacent pairs are (A, C) and (B, D). Thus, there are exactly two diagonals: AC and BD.
The statement claims there are four diagonals, which is incorrect.
Truth Value of $v$: F
Competitive Exam Focus: Truth Values
For competitive exams, you need to be quick and accurate in determining the truth value of a given statement. This often involves:
- Knowing fundamental mathematical definitions (e.g., prime numbers, rational/irrational numbers, geometric shapes).
- Understanding basic arithmetic and algebraic properties.
- Identifying counterexamples if a statement claims something is true "for all" cases.
- Recognizing that even if you don't immediately know the truth value (like complex mathematical conjectures), if the statement is unambiguous and declarative, it still *has* a definite truth value (True or False) and is therefore a statement. Questions about truth values usually involve facts expected to be known at the exam level.
Assigning truth values is the basis for evaluating compound statements and logical arguments, which are common topics in logical reasoning sections.