Session 7D
Application mini-proof set
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- At this stage, you should be comfortable choosing a proof method rather than defaulting to one. If a statement is universal, a direct proof or case proof often works. If it's an implication with a hard direction, try the contrapositive. If it asserts nonexistence or minimality, contradiction is often clean. If it involves sets, do element-chasing. If it involves "same remainder," translate to divisibility and use modular arithmetic. If it involves "for all n," consider induction if there's a natural n to n+1 connection.
- The skill you are building is proof strategy. Many beginner proofs fail because the writer does not decide what they are allowed to assume and what they must show. A good proof begins by restating the definitions and isolating the goal. Once you see the goal in definitional form, the path becomes much clearer. This is why learning the definitions and practicing rewriting is more important than memorizing tricks.
Practice
- 1If n == m (mod 5) then n^2 == m^2 (mod 5).
- 2If gcd(a,b) = 1 and a | b then a = 1 or a = -1.
- 3n(n+1)(n+2) divisible by 3.
- 4If f is bijective, then f has an inverse function.
Week quiz
- 1Show 'same remainder mod 4' is an equivalence relation, and list the 4 classes.
Show quiz answers
Quiz answers
- 1Classes: [0], [1], [2], [3] mod 4 (integers with those remainders).
MCQ
If f is bijective, then: