Logic, Conditionals, and Biconditionals
A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.
A conditional statement has the form 'if p, then q' where p is the hypothesis and q is the conclusion. Its converse is 'if q, then p'; its inverse is 'if not p, then not q'; its contrapositive is 'if not q, then not p'. A conditional and its contrapositive are always logically equivalent. A biconditional 'p if and only if q' is true exactly when both the conditional and its converse are true.
What you'll learn
- Identify the hypothesis and conclusion of a conditional statement
- Write the converse, inverse, and contrapositive of a conditional
- Recognize biconditionals and when they apply
Worked example
Problem. Write the converse of 'If a figure is a square, then it has four sides.'
- Swap the hypothesis ('a figure is a square') and the conclusion ('it has four sides').
- Converse: 'If a figure has four sides, then it is a square.'
- Note: this converse is FALSE — a rectangle has four sides too.
Answer: If a figure has four sides, then it is a square.
Practice problems
1. In 'If it rains, then the ground is wet,' what is the hypothesis?
Choices: It rains · The ground is wet
Show solution
- Warm-up: First identify exactly what the question is asking: In 'If it rains, then the ground is wet,' what is the hypothesis?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The 'if' clause is the hypothesis.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: It rains
2. In the same statement, what is the conclusion?
Choices: It rains · The ground is wet
Show solution
- Warm-up: First identify exactly what the question is asking: In the same statement, what is the conclusion?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The 'then' clause is the conclusion.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: The ground is wet
3. What is the converse of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Warm-up: First identify exactly what the question is asking: What is the converse of 'If p, then q'?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Converse swaps hypothesis and conclusion.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: If q, then p
4. What is the inverse of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Core Practice: First identify exactly what the question is asking: What is the inverse of 'If p, then q'?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Inverse negates both parts but keeps the order.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: If not p, then not q
5. What is the contrapositive of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Core Practice: First identify exactly what the question is asking: What is the contrapositive of 'If p, then q'?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Contrapositive negates AND swaps.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: If not q, then not p
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