Inverse Statements
A free Logic lesson from the “Conditionals” unit, with a worked example and practice problems including step-by-step solutions.
The inverse negates both parts without switching them. Like the converse, it is not automatically equivalent to the original conditional. Learning objective: Write and test the inverse of a conditional. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: In 'If a number is divisible by 4, then it is even,' the hypothesis is 'divisible by 4.' Example 2: The contrapositive is 'If a number is not even, then it is not divisible by 4.' A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Write and test the inverse of a conditional
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Inverse Statements): Worked example: For "If a number is divisible by 4, then the number is even.", choose the inverse.
- Worked Example: First identify exactly what the question is asking: Example case A (Inverse Statements): Worked example: For "If a number is divisible by 4, then the number is even.", choose the inverse.
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
Answer: If a number is not divisible by 4, then the number is not even.
Practice problems
1. Practice case A (Inverse Statements): What is the inverse of "If a number is divisible by 4, then the number is even."?
Choices: If a number is not divisible by 4, then the number is not even. · If the number is even, then a number is divisible by 4. · If the number is not even, then a number is not divisible by 4. · a number is divisible by 4 and the number is even.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Inverse Statements): What is the inverse of "If a number is divisible by 4, then the number is even."?
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
- So p -> q becomes ¬p -> ¬q.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If a number is not divisible by 4, then the number is not even.
2. Practice case B (Inverse Statements): What is the inverse of "If a figure is a square, then the figure is a rectangle."?
Choices: If a figure is not a square, then the figure is not a rectangle. · If the figure is a rectangle, then a figure is a square. · If the figure is not a rectangle, then a figure is not a square. · a figure is a square and the figure is a rectangle.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case B (Inverse Statements): What is the inverse of "If a figure is a square, then the figure is a rectangle."?
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
- So p -> q becomes ¬p -> ¬q.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If a figure is not a square, then the figure is not a rectangle.
3. Practice case C (Inverse Statements): What is the inverse of "If a student scores at least 70, then the quiz is passed."?
Choices: If a student does not score at least 70, then the quiz is not passed. · If the quiz is passed, then a student scores at least 70. · If the quiz is not passed, then a student does not score at least 70. · a student scores at least 70 and the quiz is passed.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Inverse Statements): What is the inverse of "If a student scores at least 70, then the quiz is passed."?
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
- So p -> q becomes ¬p -> ¬q.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If a student does not score at least 70, then the quiz is not passed.
4. Practice case D (Inverse Statements): What is the inverse of "If a triangle is equilateral, then the triangle is isosceles."?
Choices: If a triangle is not equilateral, then the triangle is not isosceles. · If the triangle is isosceles, then a triangle is equilateral. · If the triangle is not isosceles, then a triangle is not equilateral. · a triangle is equilateral and the triangle is isosceles.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case D (Inverse Statements): What is the inverse of "If a triangle is equilateral, then the triangle is isosceles."?
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
- So p -> q becomes ¬p -> ¬q.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If a triangle is not equilateral, then the triangle is not isosceles.
5. Practice case E (Inverse Statements): What is the inverse of "If a number is divisible by 4, then the number is even."?
Choices: If a number is not divisible by 4, then the number is not even. · If the number is even, then a number is divisible by 4. · If the number is not even, then a number is not divisible by 4. · a number is divisible by 4 and the number is even.
Show solution
- Core Practice: First identify exactly what the question is asking: Practice case E (Inverse Statements): What is the inverse of "If a number is divisible by 4, then the number is even."?
- For an inverse, keep the same order but negate both the hypothesis and the conclusion.
- The inverse keeps the original order.
- It negates both the hypothesis and the conclusion.
- So p -> q becomes ¬p -> ¬q.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If a number is not divisible by 4, then the number is not even.
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