Introduction to Contradiction
A free Logic lesson from the “Proof Readiness” unit, with a worked example and practice problems including step-by-step solutions.
Temporarily assume the exact negation of the desired claim. With the fixed premises, derive a consequence that cannot hold. Identify the impossibility and reject the temporary assumption. For example, if a greatest integer N existed, the integer N+1 would be greater, contradicting greatest.
What you'll learn
- Understand the basic idea of assuming the opposite to expose an impossibility
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. To prove there is no greatest integer by contradiction, what assumption starts the proof?
- Assume the negation of the desired nonexistence claim.
Answer: There is a greatest integer N
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. To prove there is no greatest integer by contradiction, what assumption starts the proof?
Choices: There is no greatest integer · Every integer is positive · N is even · There is a greatest integer N
Show solution
- Assume the negation of the desired nonexistence claim.
Answer: There is a greatest integer N
2. Assume N is the greatest integer. Which construction contradicts that assumption?
Choices: N=N · 2N is larger for every integer N · N+1 is an integer larger than N · N-1 is an integer
Show solution
- Integers are closed under adding 1.
- N+1 exceeds N regardless of its sign.
Answer: N+1 is an integer larger than N
3. Assume a smallest positive real number r exists. Which positive real number contradicts minimality?
Choices: -r · r/2 · 2r · 0
Show solution
- Since r>0, r/2 is positive and strictly smaller than r.
- Zero and negative values would not belong to the required set.
Answer: r/2
4. To prove an odd integer cannot equal an even integer, suppose 2a+1=2b with integers a,b. Which impossible consequence follows?
Choices: 1=2(b-a), so an odd number would be even · a and b are integers · 2a is even · b-a is an integer
Show solution
- Subtract 2a from both sides.
- An integer multiple of 2 cannot equal 1.
Answer: 1=2(b-a), so an odd number would be even
5. A student assumes the desired conclusion and derives a true identity. Has this proved the conclusion by contradiction?
Choices: Yes; any true identity proves the assumption · Yes; the conclusion can always be assumed · No; contradiction never uses assumptions · No; assume its negation and derive an impossibility
Show solution
- A consistent consequence of the desired claim is not a proof of that claim.
Answer: No; assume its negation and derive an impossibility
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Write a contradiction proof
Prove that there is no smallest positive real number. Explain why zero is not a valid substitute for your constructed number.
Review rubric and solution
- Assume a smallest positive real r exists.
- Construct r/2 and verify 0<r/2<r.
- Identify the contradiction and reject the assumption.
- Explain that zero is outside the positive domain.
- Assume r is the smallest positive real.
- Since r>0, 0<r/2<r, contradicting the minimality of r.
- Thus no smallest positive real exists. Zero could not contradict minimality because it is not positive.
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