Session 2B
If and only if (iff)
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
- A statement of the form P iff Q (P <-> Q) means two implications at once: P -> Q and Q -> P. This is why "iff" proofs always have two directions, usually labeled "(->)" and "(<-)" or "(1)" and "(2)." The mindset is: you are proving equivalence, so you must show each condition guarantees the other. If you only prove one direction, you have not proved "iff," you have proved only a one-way implication.
- Good "iff" proofs lean heavily on definitions and known lemmas. For example, "n is even iff n^2 is even" is proven by two short pieces: (->) even implies square even by substituting n = 2k, and (<-) square even implies even by a contrapositive or a separate lemma. Many math definitions are iff statements, like "n is odd iff n = 2k + 1 for some integer k." Treating "iff" properly also prevents logical errors in algebraic manipulation, where people assume they can reverse steps that are not reversible.
Worked examples
- An integer n is even iff n^2 is even (prove both directions).
Practice
- 1Prove: n is odd iff n^2 is odd.
- 2Prove: n is divisible by 5 iff its last digit is 0 or 5 (or prove the easy direction).
Show answers / outlines
Answers
- 1Mirror the even proof for both directions.
- 2Use n = 10k + d and analyze d in {0,...,9}.
MCQ
To prove P iff Q, what must you show?