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Statements

3.1 Statements | Mathematics Form 4
MATHEMATICS • Form 4 • Chapter 3: Logical Reasoning

3.1 Statements

In mathematical logic, communication must be clear, exact, and free from ambiguity. Not every sentence qualifies as a mathematical statement. A statement must possess an indisputable truth value—it must either be completely true or completely false, but never both at the same time.

Learning Objectives

  • Explain the meaning of a statement and determine its truth value.
  • Negate a statement using "no" or "not" (\(\sim p\)) and determine the truth value of the negation.
  • Determine the truth value of a compound statement formed by combining two statements using "and" or "or".
  • Construct statements in the form of implications: "if \(p\), then \(q\)" and "\(p\) if and only if \(q\)".
  • Construct and compare the truth values of the converse, inverse, and contrapositive of an implication.
  • Determine a counter-example to negate or disprove the truth of a particular statement.

Truth Value Matrix of Related Implications

See how an implication and its contrapositive share the same truth value, while the converse and inverse share the same truth value.

\(p\) \(q\) Statement: If \(p\), then \(q\) Converse: If \(q\), then \(p\) Inverse: If \(\sim p\), then \(\sim q\) Contrapositive: If \(\sim q\), then \(\sim p\)
True True True True True True
True False False True True False
False True True False False True
False False True True True True
\(p \rightarrow q\) ⇔ \(\sim q \rightarrow \sim p\) \(q \rightarrow p\) ⇔ \(\sim p \rightarrow \sim q\)

Brief Explanation

What is a Statement?

A statement is a sentence of which the truth value can be determined, that is either true or false, but not both.

What is a Truth Value?

The truth value of a statement is its classification as either true or false.

\(p\) \(q\) \(p \text{ and } q\) \(p \text{ or } q\)
True True True True
True False False True
False True False True
False False False False
1
Identify the statement
Check whether the sentence has a definite truth value. Questions, commands and open sentences are not statements.
2
Determine the truth value
Classify the statement as true or false. For example, \(28 + 12 = 40\) is true, while \(3^2 + 4^2 = 7^2\) is false.
3
Negate if needed
Insert "no" or "not" to form \(\sim p\). The negation reverses the truth value.
4
Combine or form implications
Use "and" or "or" for compound statements, or form "if \(p\), then \(q\)" and "\(p\) if and only if \(q\)".

Try Answering First

Read the question, answer it in your mind, then press "Check Answer".

1

Determine whether the sentence "\(-6 < -8\)" is a statement or not a statement, and state its truth value.

Answer: It is a statement because its truth value can be determined; it is a false statement (since \(-6\) is greater than \(-8\)).
2

State the antecedent and consequent for the implication: "If \(x\) is a factor of \(16\), then \(x\) is a factor of \(64\)."

Answer: Antecedent: \(x\) is a factor of \(16\); Consequent: \(x\) is a factor of \(64\).
3

Given the statement "If \(x = 5\), then \(x + 1 = 6\)", write its contrapositive.

Answer: If \(x + 1 \neq 6\), then \(x \neq 5\).

Common Mistakes

  • ! Confusing commands or open equations (like \(3x + 5 = -7\)) with statements. Without specifying the domain or value of \(x\), an equation with an unknown variable is an open sentence, not a statement.
  • ! Assuming that the converse of a true implication is always true. For example, "If \(x > 5\), then \(x > 3\)" is true, but its converse "If \(x > 3\), then \(x > 5\)" is false (e.g., \(x = 4\)).
  • ! Misidentifying the truth value of "If \(p\), then \(q\)" when the antecedent \(p\) is false. In mathematical logic, if \(p\) is false, the implication is automatically true.
  • ! Forgetting that "or" in mathematics is inclusive; a statement \(p \text{ or } q\) is true when both \(p\) and \(q\) are true.

Misconceptions

Misconception

A false statement is "not a statement".

Correct Fact

A statement does not have to be true; it simply must have a definitive truth value (either true or false).

Misconception

The inverse of an implication is formed by simply swapping antecedent and consequent.

Correct Fact

Swapping creates the converse; negating both creates the inverse; doing both creates the contrapositive.

Misconception

A counter-example requires proving all cases false.

Correct Fact

A single valid counter-example is sufficient to disprove a universal claim.

Summary

  • A statement is a sentence that is either true or false, but not both.
  • The negation of statement \(p\) is written as \(\sim p\) and has the opposite truth value.
  • A compound statement with "and" is true only when all component statements are true.
  • A compound statement with "or" is true if at least one component statement is true.
  • In an implication "If \(p\), then \(q\)", \(p\) is the antecedent and \(q\) is the consequent.
  • "\(p\) if and only if \(q\)" represents two implications: "If \(p\), then \(q\)" and "If \(q\), then \(p\)".
  • For an implication: Converse is "If \(q\), then \(p\)"; Inverse is "If \(\sim p\), then \(\sim q\)"; Contrapositive is "If \(\sim q\), then \(\sim p\)".
  • Equivalent pairs: An implication shares the same truth value with its contrapositive; the converse shares the same truth value with the inverse.
  • A counter-example is an instance that disproves a false general statement.

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 Which of the following sentences is a statement?

2 Which related implication always has the same truth value as the conditional statement "If \(p\), then \(q\)"?

B. Fill in the Blanks

3 In the implication "If \(k\) is divisible by 5, then \(k\) is a multiple of 5", the phrase "\(k\) is divisible by 5" is known as the .

4 If statement \(p\) is false and statement \(q\) is true, the compound statement "\(p \text{ or } q\)" has a truth value of .

C. Drag and Drop: Match Each Logical Form

Match each logical form of an implication based on statement \(p\) and statement \(q\) to its correct definition. On a phone, tap the answer choice first, then tap the definition box.

Logical Form
Statement
Converse
Inverse
Contrapositive
Structure / Definition
If \(p\), then \(q\).CorrectTry againEmpty
If \(q\), then \(p\).CorrectTry againEmpty
If \(\sim p\), then \(\sim q\).CorrectTry againEmpty
If \(\sim q\), then \(\sim p\).CorrectTry againEmpty
 

Key Terms

Statement Truth Value Negation (\(\sim p\)) Compound Statement Implication Antecedent Consequent Converse Inverse Contrapositive Counter-example
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.

Statements

3.1 Statements | Mathematics Form 4
MATHEMATICS • Form 4 • Chapter 3: Logical Reasoning

3.1 Statements

In mathematical logic, communication must be clear, exact, and free from ambiguity. Not every sentence qualifies as a mathematical statement. A statement must possess an indisputable truth value—it must either be completely true or completely false, but never both at the same time.

Learning Objectives

  • Explain the meaning of a statement and determine its truth value.
  • Negate a statement using "no" or "not" (\(\sim p\)) and determine the truth value of the negation.
  • Determine the truth value of a compound statement formed by combining two statements using "and" or "or".
  • Construct statements in the form of implications: "if \(p\), then \(q\)" and "\(p\) if and only if \(q\)".
  • Construct and compare the truth values of the converse, inverse, and contrapositive of an implication.
  • Determine a counter-example to negate or disprove the truth of a particular statement.

Truth Value Matrix of Related Implications

See how an implication and its contrapositive share the same truth value, while the converse and inverse share the same truth value.

\(p\) \(q\) Statement: If \(p\), then \(q\) Converse: If \(q\), then \(p\) Inverse: If \(\sim p\), then \(\sim q\) Contrapositive: If \(\sim q\), then \(\sim p\)
True True True True True True
True False False True True False
False True True False False True
False False True True True True
\(p \rightarrow q\) ⇔ \(\sim q \rightarrow \sim p\) \(q \rightarrow p\) ⇔ \(\sim p \rightarrow \sim q\)

Brief Explanation

What is a Statement?

A statement is a sentence of which the truth value can be determined, that is either true or false, but not both.

What is a Truth Value?

The truth value of a statement is its classification as either true or false.

\(p\) \(q\) \(p \text{ and } q\) \(p \text{ or } q\)
True True True True
True False False True
False True False True
False False False False
1
Identify the statement
Check whether the sentence has a definite truth value. Questions, commands and open sentences are not statements.
2
Determine the truth value
Classify the statement as true or false. For example, \(28 + 12 = 40\) is true, while \(3^2 + 4^2 = 7^2\) is false.
3
Negate if needed
Insert "no" or "not" to form \(\sim p\). The negation reverses the truth value.
4
Combine or form implications
Use "and" or "or" for compound statements, or form "if \(p\), then \(q\)" and "\(p\) if and only if \(q\)".

Try Answering First

Read the question, answer it in your mind, then press "Check Answer".

1

Determine whether the sentence "\(-6 < -8\)" is a statement or not a statement, and state its truth value.

Answer: It is a statement because its truth value can be determined; it is a false statement (since \(-6\) is greater than \(-8\)).
2

State the antecedent and consequent for the implication: "If \(x\) is a factor of \(16\), then \(x\) is a factor of \(64\)."

Answer: Antecedent: \(x\) is a factor of \(16\); Consequent: \(x\) is a factor of \(64\).
3

Given the statement "If \(x = 5\), then \(x + 1 = 6\)", write its contrapositive.

Answer: If \(x + 1 \neq 6\), then \(x \neq 5\).

Common Mistakes

  • ! Confusing commands or open equations (like \(3x + 5 = -7\)) with statements. Without specifying the domain or value of \(x\), an equation with an unknown variable is an open sentence, not a statement.
  • ! Assuming that the converse of a true implication is always true. For example, "If \(x > 5\), then \(x > 3\)" is true, but its converse "If \(x > 3\), then \(x > 5\)" is false (e.g., \(x = 4\)).
  • ! Misidentifying the truth value of "If \(p\), then \(q\)" when the antecedent \(p\) is false. In mathematical logic, if \(p\) is false, the implication is automatically true.
  • ! Forgetting that "or" in mathematics is inclusive; a statement \(p \text{ or } q\) is true when both \(p\) and \(q\) are true.

Misconceptions

Misconception

A false statement is "not a statement".

Correct Fact

A statement does not have to be true; it simply must have a definitive truth value (either true or false).

Misconception

The inverse of an implication is formed by simply swapping antecedent and consequent.

Correct Fact

Swapping creates the converse; negating both creates the inverse; doing both creates the contrapositive.

Misconception

A counter-example requires proving all cases false.

Correct Fact

A single valid counter-example is sufficient to disprove a universal claim.

Summary

  • A statement is a sentence that is either true or false, but not both.
  • The negation of statement \(p\) is written as \(\sim p\) and has the opposite truth value.
  • A compound statement with "and" is true only when all component statements are true.
  • A compound statement with "or" is true if at least one component statement is true.
  • In an implication "If \(p\), then \(q\)", \(p\) is the antecedent and \(q\) is the consequent.
  • "\(p\) if and only if \(q\)" represents two implications: "If \(p\), then \(q\)" and "If \(q\), then \(p\)".
  • For an implication: Converse is "If \(q\), then \(p\)"; Inverse is "If \(\sim p\), then \(\sim q\)"; Contrapositive is "If \(\sim q\), then \(\sim p\)".
  • Equivalent pairs: An implication shares the same truth value with its contrapositive; the converse shares the same truth value with the inverse.
  • A counter-example is an instance that disproves a false general statement.

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 Which of the following sentences is a statement?

2 Which related implication always has the same truth value as the conditional statement "If \(p\), then \(q\)"?

B. Fill in the Blanks

3 In the implication "If \(k\) is divisible by 5, then \(k\) is a multiple of 5", the phrase "\(k\) is divisible by 5" is known as the .

4 If statement \(p\) is false and statement \(q\) is true, the compound statement "\(p \text{ or } q\)" has a truth value of .

C. Drag and Drop: Match Each Logical Form

Match each logical form of an implication based on statement \(p\) and statement \(q\) to its correct definition. On a phone, tap the answer choice first, then tap the definition box.

Logical Form
Statement
Converse
Inverse
Contrapositive
Structure / Definition
If \(p\), then \(q\).CorrectTry againEmpty
If \(q\), then \(p\).CorrectTry againEmpty
If \(\sim p\), then \(\sim q\).CorrectTry againEmpty
If \(\sim q\), then \(\sim p\).CorrectTry againEmpty
 

Key Terms

Statement Truth Value Negation (\(\sim p\)) Compound Statement Implication Antecedent Consequent Converse Inverse Contrapositive Counter-example
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Logic Reasoning
Topic : Statement
Form 4 Mathematics
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