Session 7A
Relations
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 relation on a set A is any rule that tells you which pairs (a,b) in A x A are related. Relations generalize ideas like "<=", "same parity," and "congruent mod n." Relations can have properties: reflexive means every element relates to itself (aRa). Symmetric means if aRb then bRa. Transitive means if aRb and bRc then aRc. Proofs here are direct definition proofs: take the definition, assume the hypothesis, and show the property.
- A standard beginner example is the relation "aRb if a-b is even" on integers. Reflexive holds because a-a=0 is even. Symmetric holds because if a-b is even, then b-a = -(a-b) is also even. Transitive holds because if a-b and b-c are even, then adding gives a-c = (a-b) + (b-c), which is even. These proofs train you to do "definition, then algebra" cleanly, which is the same structure you use later in abstract algebra.
Practice
On integers, define a R b if a - b is even. Prove R is reflexive.
- 1Prove R is symmetric.
- 2Prove R is transitive.
MCQ
For a R b defined by a - b even, which property follows from 'if a - b is even then b - a is even'?