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 |
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 |
Try Answering First
Read the question, answer it in your mind, then press "Check Answer".
Determine whether the sentence "\(-6 < -8\)" is a statement or not a statement, and state its truth value.
State the antecedent and consequent for the implication: "If \(x\) is a factor of \(16\), then \(x\) is a factor of \(64\)."
Given the statement "If \(x = 5\), then \(x + 1 = 6\)", write its contrapositive.
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
A false statement is "not a statement".
A statement does not have to be true; it simply must have a definitive truth value (either true or false).
The inverse of an implication is formed by simply swapping antecedent and consequent.
Swapping creates the converse; negating both creates the inverse; doing both creates the contrapositive.
A counter-example requires proving all cases false.
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.
