Session 8B
Mixed proof exam (practice)
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
- Mixed practice is where you learn to select the right tool. If you see an "iff," split into two directions. If you see set equality, prove two inclusions. If you see a divisibility statement, rewrite using "= ak." If you see something about squares and parity, consider cases mod 2 or mod 4. If you see a sum or a power indexed by n, consider induction. Strategy becomes almost mechanical once you recognize these signal phrases and structures.
- The goal is not to know advanced theorems. It's to know how to translate a problem into the right language and then apply a standard proof pattern cleanly. This is why earlier weeks drilled translation, negation, and definitions. In a mixed proof session, you should always start by rewriting the statement in precise math form, then pick a method, then write the proof with explicit steps. Skipping the rewrite step is the fastest way to get stuck.
Practice
- 1If n is odd then n^2 == 1 (mod 8) is false; fix it.
- 2Prove: A \ (B intersect C) = (A \ B) union (A \ C).
- 3Induction: 1^3 + 2^3 + ... + n^3 = (n(n+1)/2)^2.
- 4Prove Euclid's Lemma (outline ok).
- 5If f and g are bijections then f o g is a bijection.
- 6Pigeonhole: among any 10 integers, two have the same last digit.
MCQ
Which identity is correct?