Practice question · Sort into groups
Sort each claim by what it would take to settle it.
Groups: Settled by one example · Settled by one counterexample · Needs a general proof
- The sum of two odd numbers is even
- Some even number is prime
- Every prime is odd
- There is an integer between 3 and 5
Hints
- Existential claims fall to a single witness; false universals fall to a single counterexample.
- A universal claim that is TRUE cannot be established by examples alone.
Show the answer
Settled by one example: Some even number is prime, There is an integer between 3 and 5
Settled by one counterexample: Every prime is odd
Needs a general proof: The sum of two odd numbers is even
Why
Existential claims (p1, p4) need one witness. A false universal (p2) needs one counterexample. But a true universal (p3) covers infinitely many cases, so no finite pile of examples proves it, only a general argument does.
Practise Direct Proof and Counterexample
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More questions on Direct Proof and Counterexample
- Checking that a claim about all integers holds for the first ten thousand cases is not a proof, while a…
- A claim that holds for the first one million integers is thereby proven for all integers.
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- Claim: 'every prime number is odd'. Select every value that is a genuine COUNTEREXAMPLE.
- Every even number above 2 checked so far is the sum of two primes, verified past 4 × 10¹⁸. Why is Goldbach's…