Foundations path · F06
Before this lesson: Mathematical language and sets
Your goal: Write a connected argument with a reason for each decisive step.
Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.
Mathematical Proof: Direct Proof and Contradiction: the key idea
A proof covers every case allowed by the question. Examples help you discover a claim, but do not prove a universal statement. A direct proof starts from the assumptions. A proof by cases divides the domain into exhaustive possibilities. A contradiction assumes the desired conclusion fails and derives an impossibility. To find all solutions, first show that every solution lies in your list, then check that every listed candidate works. A converse reverses an implication and must be proved separately.
A worked example
Prove that the product of two odd integers is odd.
Write the integers as 2r+1 and 2s+1 with r,s integers. Their product is 4rs+2r+2s+1 = 2(2rs+r+s)+1. The bracket is an integer, so the product has the required odd form.
Your turn: change one thing
Is the converse of “divisible by 4 implies even” true?
Try this on paper before opening the explanation.
Compare your reasoning
No. The integer 6 is even but not divisible by 4. A converse needs its own argument, and one counterexample defeats this one.
Pause and check
A trap to avoid: State the domain, keep exact values and explain why each step is allowed. A correct answer still needs a reason.
Practise and adjust the level
Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.
Interactive practice loads here. You can also use the complete question set below.
Write a complete argument
Prove that there is no smallest positive rational number.
Planning hint
List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.
Read the full solution after your attempt
Suppose r were the smallest positive rational. Then r/2 is rational, positive and smaller than r. This contradicts the supposed minimality. Since the construction works for any proposed r, no smallest positive rational exists.
My proof notebook
Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.
All nine practice questions
Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.
1. What does a proof of a universal statement need to cover?
Foundation
- A typical diagram
- Every permitted case
- Ten examples
- Only positive inputs regardless of the question
Hint
Read the quantifier and domain.
Answer and reasoning
Every permitted case. A universal claim applies to every member of its stated domain.
2. Which is a counterexample to “all even integers are divisible by 4”?
Foundation
- 8
- 12
- 16
- 6
Hint
Find an even integer leaving remainder 2 modulo 4.
Answer and reasoning
6. 6 is even, but 6/4 is not an integer.
3. Which cases cover every integer without overlap?
Core
- Prime and even
- Square and cube
- Even and odd
- Positive and even
Hint
A case split must be exhaustive.
Answer and reasoning
Even and odd. Every integer has remainder 0 or 1 on division by 2, exactly one of these.
4. To prove A implies B by contradiction, what do we assume?
Core
- A is false
- B is true
- A and B are both false
- A is true and B is false
Hint
Assume the hypotheses and failure of the target.
Answer and reasoning
A is true and B is false. A counterexample to A⇒B has A true and B false; ruling that out proves the implication.
5. You derived three candidates for a find-all problem. What remains?
Stretch
- Check all candidates and justify no others were lost
- Choose the nicest candidate
- Give decimals
- Stop immediately
Hint
Necessity and sufficiency are different directions.
Answer and reasoning
Check all candidates and justify no others were lost. Derivation restricts possible solutions; checking proves the candidates really satisfy the original conditions.
6. Why is checking n=1,2,3 insufficient to prove a claim for all positive integers?
Stretch
- Three is never prime
- Small numbers cannot be used in mathematics
- The claim must be false
- Later integers have not been covered
Hint
A finite check does not control every later case.
Answer and reasoning
Later integers have not been covered. A later counterexample may exist. A general argument such as induction is needed.
7. A counterexample is used to do what?
Foundation
- Prove uniqueness automatically
- Disprove a universal statement
- Prove every universal statement
- Replace all reasoning
Hint
It is a permitted input where the claim fails.
Answer and reasoning
Disprove a universal statement. One failure suffices to refute a claim about every input.
8. Which number disproves “every odd integer is prime”?
Core
- 9
- 3
- 5
- 7
Hint
Find an odd composite.
Answer and reasoning
9. 9 is odd but equals 3·3.
9. In a proof by contradiction, what is assumed?
Stretch
- That every statement is true
- The hypotheses and the negation of the conclusion
- The desired conclusion only
- An unrelated false claim
Hint
The contradiction must arise from the denied conclusion.
Answer and reasoning
The hypotheses and the negation of the conclusion. Showing these assumptions cannot all hold establishes the original conclusion.
Choose your next step
Continue to Geometric language and ratios. If this felt difficult, return to a prerequisite above. Every lesson stays open.
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