Complete or critique a solution

Welcome to the World of Mathematical Logic!

In H3 Mathematics, you aren't just a "calculator"—you are a mathematical critic and architect. One of the most important skills in the Mathematical Investigation and Reading Mathematical Texts section is the ability to complete or critique a solution.

Instead of just solving for \(x\), you will be looking at a finished (or half-finished) piece of work and asking: "Does this actually make sense?" or "What is missing to make this bridge solid?" Don't worry if this seems a bit abstract at first. Think of it like being a judge on a talent show—you're looking for the strengths and pointing out the slips!

1. Completing a Solution: The "Missing Bridge" Approach

Sometimes, a mathematician provides a "sketch" of a proof. Your job is to fill in the logical gaps to make it a complete solution. This often happens in proofs by Mathematical Induction or Direct Proof.

How to Fill the Gaps:

1. Identify the Destination: What is the final Theorem or Proposition the solution is trying to reach?
2. Check the Connectives: Are there "implies" (\(\Rightarrow\)) arrows missing? Does the next step actually follow from the previous one?
3. Check for Cases: Did the author solve for \(n > 0\) but forget the \(n = 0\) case? Completing a solution often means adding these special cases.

Example: If a proof shows that \(n^2\) is even when \(n\) is even, but the goal is to prove it for all integers, you must complete the solution by also considering the case where \(n\) is odd.

Quick Tip: Look for the word "Similarly..." in a text. Often, a mathematician uses this to skip steps. To complete the solution, you should be able to explain exactly what that "similarly" entails!

2. Critiquing a Solution: Being a Math Detective

To critique a solution means to evaluate its validity. Is the logic sound, or is there a "leak" in the argument? Even if the final answer is correct, the reasoning might be wrong.

Common "Red Flags" to Look For:

A. The Converse Error: Just because "If \(P\), then \(Q\)" is true, it doesn't mean "If \(Q\), then \(P\)" is true. Analogy: "If it is raining, the grass is wet" is true. But if you see wet grass, you cannot assume it rained—maybe someone used a sprinkler!

B. Misusing Quantifiers: Watch out for the difference between "For all" (\(\forall\)) and "There exists" (\(\exists\)). Common mistake: Proving something works for one number (\(\exists\)) and claiming it works for all numbers (\(\forall\)).

C. Dividing by Zero: This sounds simple, but in complex algebraic proofs, authors often divide by a variable (like \(x - a\)) without ensuring that \(x \neq a\). This is a classic "hole" in a critique task.

D. Circular Reasoning: Does the proof assume the conclusion is true in order to prove the conclusion? Example: To prove \(A\) is true, the author uses property \(B\). But property \(B\) only exists if \(A\) is already true. This is a logical "no-exit" loop!

Key Takeaway:

A critique isn't just saying "this is wrong." A good critique identifies where the logic fails and why (e.g., "The author assumed the converse was true without proof").

3. Logical Connectives and Conditionals

To critique effectively, you must master the language of the syllabus. If you find these confusing, think of them as "logical traffic signs."

• Necessary Condition: A condition that must be true for the conclusion to happen. (If it's not met, the conclusion is definitely false).
• Sufficient Condition: A condition that, if met, guarantees the conclusion.
• If and only if (iff): This means the relationship works perfectly in both directions. This is the "gold standard" of mathematical equivalence.

Did you know? Many students lose marks because they provide a necessary condition when the question asked for a sufficient one. Always check if the "arrow" of logic points one way or both ways!

4. Step-by-Step Guide to Critiquing a Proof

When you are handed a text to critique, follow these steps:

1. Test with a Counterexample: Before reading the logic, try a simple number. If the "solution" claims something is true for all \(n\), try \(n=0\) or \(n=1\). If it fails, you've found a Disproof by Counterexample immediately!
2. Identify the "If... then...": Clearly mark what the author is assuming (the Premise) and what they are trying to show (the Conclusion).
3. Check the Definitions: Does the author use a Definition correctly? For example, if they talk about "prime numbers," did they accidentally include 1?
4. Look for Gaps: Are there sentences like "It is obvious that..." or "Clearly..."? These are often where the weakest links in the logic hide.

5. Summary and Quick Review

Don't worry if this seems tricky at first! Critiquing is a higher-level skill that improves with practice. You are moving from being a "user" of math to an "analyst" of math.

Quick Review Box:
- Complete a solution: Find the missing steps, cases, or logical links.
- Critique a solution: Look for logical fallacies (converse error, circular reasoning, division by zero).
- Quantifiers: Ensure \(\forall\) (for all) and \(\exists\) (there exists) are used correctly.
- Counterexample: The fastest way to "break" a bad solution.

Memory Aid: Use the "C.A.S.E." method when critiquing:
Converse errors?
Assumptions (are they valid)?
Special cases (is \(n=0\) or \(n=1\) missing)?
Equivalence (is "if" being confused with "if and only if")?

Keep practicing these investigations! The more you read other people's math, the better your own mathematical writing will become.