2.2 Inductive, Deductive, and Counterexample Reasoning

From Pattern to Proof

The Logical Reasoning part of Math 30-2 expects you to move between three habits: observe a pattern, decide whether the pattern is guaranteed, and communicate why. Alberta Education's 2025-2026 bulletin describes problem solving as the ability to adapt mathematics to unfamiliar situations and verify or interpret results. That is exactly what separates a strong logic answer from a lucky guess.

Inductive reasoning uses specific observations to form a general conjecture. Deductive reasoning starts with accepted facts, definitions, or rules and applies them to a case. A counterexample is one example that makes a general claim false.

How the three tools fit together

A diploma question may show a table, geometric pattern, game scoreboard, or set of number statements and ask for a conjecture. That is inductive. Another question may give rules for moving tokens or sorting numbers and ask which conclusion must follow. That is deductive. A third question may ask which value disproves a claim. That is counterexample reasoning.

Inductive reasoning: useful but limited

Suppose a pattern of tile designs has 4 tiles in Figure 1, 7 tiles in Figure 2, 10 tiles in Figure 3, and 13 tiles in Figure 4. You may conjecture that Figure n has 3n + 1 tiles. That conjecture fits the observed data. It is still not proof unless the pattern's construction rule is known or a deductive argument shows why three tiles are added every time.

In Math 30-2, an inductive answer should say what was observed. Instead of writing "the answer is 31," write "the tile count increases by 3 each figure, so I modeled the count with 3n + 1; for Figure 10, this gives 31." The second response tells the marker and the reader how the pattern was chosen.

Deductive reasoning: rule first

Deductive reasoning is strongest when the rule is stated before the result. For example:

• Rule: All cards in set A have exactly one symbol.
• Fact: Card 18 is in set A.
• Conclusion: Card 18 has exactly one symbol.

The conclusion must follow if the rule and fact are true. Notice that the argument does not require checking every card. Deduction avoids the weakness of induction by leaning on a general rule that has already been accepted.

A common error is applying a true rule outside its conditions. If the rule says "all cards in set A have exactly one symbol" and a card has exactly one symbol, you cannot conclude that it is in set A. The rule does not say that set A contains every one-symbol card.

Counterexamples: one clean disproof

Counterexamples target claims that are supposed to work for all cases. The counterexample must meet the condition in the claim but fail the conclusion.

Claim: "All two-digit numbers divisible by 3 are odd."

A strong counterexample is 12. It is a two-digit number and it is divisible by 3, but it is not odd. The example directly enters the claim's condition and breaks the claimed result.

A weak response would be 8. It is even, but it is not divisible by 3, so it never tests the claim. Another weak response would be 123, because it is not a two-digit number. A counterexample is not just a different-looking case; it must be inside the original fence.

Worked diploma-style example

A class tests a conjecture about a puzzle game: "Whenever a player starts on an even-numbered square, the player finishes on a shaded square." The data show these starts and finishes:

The first three rows support the conjecture inductively. The fourth row disproves it. Start square 8 is even, so it satisfies the condition, but its finish is unshaded, so the conclusion fails. The correct written explanation should name both parts: "8 is an even starting square and it finishes unshaded, so it is a counterexample."

Choosing the right reasoning on the exam

When the prompt says make a conjecture, look for pattern evidence and describe it. When it says prove, justify, or must be true, use rules, definitions, or complete cases. When it says disprove or which example shows the conjecture is false, search for a counterexample that fits the condition and violates the conclusion.

This is especially important in numerical-response or multiple-choice items. Distractors often include values that break only part of the statement. Before choosing, underline the condition and conclusion. Test candidate counterexamples against both. If the claim is "all members of A are also in B," an item outside A is irrelevant, even if it is outside B too.

Writing quality matters

The Mathematics 30-2 exam includes written-response questions, and the bulletin emphasizes organized responses with pertinent ideas, calculations, formulas, and units when relevant. In logic, the equivalent of units is language such as therefore, because, contradicts, satisfies the condition, and fails the conclusion. Those words make the reasoning auditable.

A useful final check is: did I prove the exact claim, or did I only show examples? Did my counterexample actually meet every condition? Did I reverse an if-then rule? If you can answer those questions in your own explanation, your reasoning is much less likely to be fooled by a polished distractor.