Additional Mathematics Synthesis Guide 49: Counterexamples and Theorem Boundaries — Testing Claims, Conditions and Invalid Shortcuts

State the claim → state its conditions → test the boundary → search for a counterexample → keep only what the evidence really proves.

1. A mathematical claim has a scope

Every theorem or rule says something about a specified class of objects under specified conditions. The conclusion is not automatically valid when those conditions are changed or removed.

For example, the rule “if ab=ac and a≠0, then b=c” is valid. Removing a≠0 makes the conclusion false: if a=0, then ab=ac becomes 0=0 regardless of b and c.

2. Counterexample versus proof

A proof establishes a universal claim for every allowed case. A counterexample destroys a universal claim by producing just one allowed case where the conclusion fails.

To disprove

√(a+b)=√a+√b for all positive a,b,

take a=b=1. Then √2≠2. One case is enough because the original statement claimed all positive a,b.

3. Examples cannot prove a universal rule

Checking a formula at x=1,2,3,4 can increase confidence but does not establish it for every real x. A finite test set cannot by itself prove an infinite-domain identity.

This distinction is especially important when students infer algebraic or trigonometric identities from calculator evidence.

4. The converse may be false

A theorem of the form

P ⇒ Q

does not automatically imply the converse Q⇒P.

Example: if two lines are perpendicular and both gradients are finite non-zero values, their gradients multiply to −1. But one must still be careful when vertical lines are involved, because the ordinary finite-gradient product is not defined there.

5. Division by a variable expression has a boundary

From

x(x−3)=0,

dividing by x assumes x≠0 and destroys the valid solution x=0.

The shortcut is not always wrong. It is wrong when its hidden condition excludes a genuine branch.

6. Cancelling factors preserves an exclusion

For

(x²−1)/(x−1),

cancelling x−1 gives x+1 only for x≠1. The simplified expression is defined at x=1, but the original expression is not.

A counterexample to careless equivalence is x=1 itself: the simplified formula has value2 while the original has no value.

7. Squaring is not an equivalence unless sign is controlled

If A=B, then A²=B². But A²=B² allows A=B or A=−B.

Therefore squaring can enlarge a solution set. The original equation must be checked afterward.

For √(x+1)=x−1, squaring gives candidates x=0 and3, but the original right side must be non-negative; only x=3 survives.

8. Logarithm laws have domain conditions

Over the real numbers, logarithm arguments must be positive.

The identity

ln(ab)=lna+lnb

is used when the relevant logarithms exist in the real domain. Writing ln(−1)+ln(−1) as though the real logarithms were defined is outside that domain.

9. Trigonometric identities live on a common domain

Consider

sinx/cosx=tanx.

This is valid where cosx≠0. At x=π/2 the quotient is undefined and tanx is also undefined. The identity does not create a finite value there.

Similarly, cancellation inside a trig identity does not restore points excluded by an original denominator.

10. One numerical match does not prove a trig identity

Two different functions can agree at one angle and disagree elsewhere. Testing x=30° is therefore evidence about one point, not a proof of an identity.

Identity proof requires symbolic equivalence on the common domain.

11. f′(a)=0 does not automatically mean maximum or minimum

For f(x)=x³,

f′(x)=3x², so f′(0)=0.

Yet x=0 is not a maximum or minimum. It is a stationary inflexion.

This is a counterexample to the false claim “every stationary point is an extremum”.

12. f″(a)=0 does not prove an inflexion

For f(x)=x⁴,

f′(0)=0 and f″(0)=0,

but x=0 is a local minimum, not an inflexion.

So the condition f″=0 is not sufficient for inflexion. It merely makes the ordinary second-derivative max/min test inconclusive.

13. Equal gradients do not by themselves prove tangency

Two curves or lines can have equal gradients at different points. Tangency at x=a requires same position and same gradient at the contact point.

• same value: f(a)=g(a);
• same gradient: f′(a)=g′(a).

Equal gradient alone proves matching direction, not contact.

14. A graph sketch can suggest, but not establish, exact structure

A sketch can reveal likely roots, extrema and intersections. It is valuable for planning and checking. But an approximate drawing cannot establish an exact root, exact tangent parameter or exact equality unless supported by the mathematics.

Visual evidence and proof have different jobs.

15. A local fact need not imply a global fact

Knowing f′(2)>0 tells us that the function is increasing locally at x=2 in the derivative sense. It does not prove f′(x)>0 for every x in a large interval.

To prove increasing behaviour on an interval, derivative sign must be controlled throughout that interval.

16. A repeated root is a boundary event

For a quadratic, Δ=0 separates two-root and no-real-root regimes. The repeated-root case deserves separate treatment because it is the threshold where behaviour changes.

Careless statements such as “Δ≤0 means no real roots” are false at the boundary Δ=0.

17. Strict and non-strict inequalities have different theorem boundaries

For an upward-opening quadratic, Δ<0 supports strict positivity everywhere. Allowing Δ=0 gives non-negativity, because the graph can touch zero once.

The equality boundary cannot be ignored when the wording changes from “positive” to “non-negative”.

18. Counterexamples can be designed systematically

When testing a suspicious universal claim, choose cases that attack its weakest condition.

• Test zero if division or cancellation is involved.
• Test negative values if square roots or logs are involved.
• Test endpoints if an interval claim is made.
• Test a stationary inflexion if “stationary means extremum” is claimed.
• Test repeated roots when a statement ignores equality boundaries.
• Test a vertical line when a gradient formula assumes finite gradients.

19. The theorem-boundary checklist

1. Write the exact claim.
2. Identify whether it is universal, existential or conditional.
3. List every hypothesis: domain, non-zero condition, interval, differentiability, sign, parameter regime.
4. Ask what happens if one hypothesis is removed.
5. Search for a simple boundary case: zero, endpoint, repeated root, undefined point or symmetry point.
6. If one valid counterexample appears, reject the universal claim.
7. If no counterexample appears, do not call that a proof; build a general argument.
8. State the corrected theorem with its necessary conditions.

20. Common failure patterns

• Checking several examples and calling the pattern proved.
• Using a theorem without its domain or non-zero condition.
• Assuming a converse automatically holds.
• Dividing by a variable factor and losing a zero branch.
• Squaring and accepting every new root.
• Calling every stationary point a maximum or minimum.
• Calling f″=0 proof of inflexion.
• Calling equal gradients proof of tangency without checking the same point.
• Reading an approximate graph as exact proof.
• Ignoring equality cases at parameter thresholds.

21. Practice set

1. Give a counterexample to √(a+b)=√a+√b for all positive a,b.
2. Why do five successful numerical tests not prove a universal identity?
3. What extra condition is needed to infer b=c from ab=ac?
4. What solution is lost by dividing x(x−3)=0 by x?
5. Why does cancelling x−1 from (x²−1)/(x−1) not make x=1 valid?
6. What logical problem can squaring create?
7. What real-domain condition applies to lnx?
8. Where is tanx undefined?
9. Give a counterexample to “every stationary point is an extremum”.
10. Give a counterexample to “f″(a)=0 proves inflexion”.
11. What two conditions establish tangency at x=a?
12. Why is a sketch not enough to prove an exact root?
13. Does f′(2)>0 prove f is increasing on all real x?
14. What does Δ=0 mean for a quadratic?
15. Why is Δ≤0 not the correct condition for strict positivity of an upward quadratic?
16. What boundary value should be tested when division is involved?
17. What boundary case should be tested in parameter-root problems?
18. If a universal claim has one valid counterexample, what follows?
19. If ten tests reveal no counterexample, has the theorem been proved?
20. What should replace an invalid shortcut in a final solution?

Answers

1. a=b=1 gives √2≠2.
2. Finite tests cannot cover every member of an infinite domain.
3. a≠0.
4. x=0.
5. The original denominator is zero at x=1, so the original expression is undefined there.
6. It can create extra candidate roots because A²=B² allows A=±B.
7. x>0.
8. x=π/2+kπ.
9. f(x)=x³ at x=0.
10. f(x)=x⁴ at x=0.
11. Same value and same gradient.
12. A sketch is approximate visual evidence; exact claims need algebraic or theorem-based justification.
13. No. It is a local derivative fact at one point.
14. One repeated real root.
15. Δ=0 allows the graph to touch zero, giving non-negativity rather than strict positivity.
16. Zero.
17. The repeated-root threshold Δ=0.
18. The universal claim is false.
19. No. A general proof is still required.
20. A legal argument that states and preserves the necessary conditions.

22. What mastery looks like

Mastery means the learner reads every theorem with its hypotheses attached, distinguishes evidence from proof, can design a counterexample against an overextended claim and recognises equality or domain boundaries before they become hidden assumptions.

The transfer test is to present a plausible but false shortcut. If the learner can identify the missing condition, produce a counterexample and replace the shortcut with a valid statement, theorem-boundary control has become part of A-Math reasoning.

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Continue through Batch 13

• Guide 50: Generalisation from Cases to Families — Patterns, Parameters, Conjectures and Exact Rules
• Guide 51: Local-to-Global Reasoning — Points, Intervals, Extrema and Whole-Graph Conclusions
• Guide 52: Mathematical State Tracking — Variables, Conditions, Exactness, Branches and Intermediate Results
• Guide 39: Proof and Justification
• Guide 46: Case Splitting and Branch Control
• Additional Mathematics Directory
• BTT Mathematics Hub