Direct Proof, Cases and Implications
Follow and construct direct deductions or proofs by cases and derive valid implications from stated facts.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Build a direct proof as a sequence of justified deductions from assumptions.
Exam cue: Write the assumption and required conclusion before filling the chain.
Concept 2
Partition a problem into exhaustive cases such as even and odd.
Exam cue: Check that proof cases are exhaustive and non-overlapping where needed.
Concept 3
Combine given statements only in directions licensed by their logic.
Exam cue: Name the fact that justifies each transition.
Risk pitfalls and guardrails
Using the desired conclusion as an unstated assumption.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Checking examples instead of proving the general case.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Applying an implication backwards.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Direct chain
Start from what is known and justify every step toward the target.
Cases
Split into cases that together cover every permitted possibility.
Even form
Represent an even integer as 2k.
Odd form
Represent an odd integer as 2k+1.
Implication direction
A⇒B licenses A to B, not automatically B to A.
No circularity
A proof cannot assume the result it is meant to establish.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
To prove the sum of two even integers is even, write them as 2a and 2b. Which final form is needed?
If n=2k+1, which expression proves n² is odd?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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