One example is enough to show a statement is false. Decide whether statements are always, sometimes or never true.
What you will learn
- Disprove a statement with a counterexample
- Decide if a statement is always, sometimes or never true
- Choose the right algebra for a proof
One example breaks it
To prove "always" you need algebra.
To disprove it you need just one counterexample.
"n² + n + 41 is always prime" holds for n = 1 to 39, but n = 40 gives 1681 = 41 × 41. One value is enough.
Good places to look: 0, 1, negative numbers, fractions between 0 and 1, and 2 (the only even prime).
To find the smallest value that breaks a rule, test n = 1, 2, 3, ... in order and stop at the first one that fails. "n² + 6 is always prime": n = 1 gives 7, prime; n = 2 gives 10 = 2 × 5, not prime. So n = 2 is the smallest counterexample. With a formula like n² − n + 13, try n = 13 as well: 13² − 13 + 13 = 13 × 13, not prime.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- One example breaks it
- Always, sometimes or never
- Counterexamples to look for
- Proving instead
7 practice questions, marked as you go; about 20 minutes in all.
Key terms
Counterexample
In this unit: Proof and iteration
Algebraic proof and counterexamples, then finding roots that do not come out exactly: changes of sign and iteration. Higher tier.
- Algebraic proof
- Counterexamples
- Roots by a change of sign
- Iteration