Practice question · Put in order
Order the steps that verify 'has the same parity as' is an equivalence relation.
- All three properties hold, so the relation is an equivalence relation
- Reflexive: every integer has the same parity as itself
- Symmetric: if a and b share a parity, then b and a do too
- Transitive: if a matches b and b matches c, then a matches c
Hints
- The three properties are checked in their standard order before concluding.
- The conclusion comes only after all three are established.
Show the answer
- Reflexive: every integer has the same parity as itself
- Symmetric: if a and b share a parity, then b and a do too
- Transitive: if a matches b and b matches c, then a matches c
- All three properties hold, so the relation is an equivalence relation
Why
Verification is mechanical: check reflexivity, symmetry and transitivity in turn, then conclude. Skipping any one of them leaves the conclusion unearned, and it is transitivity that most often fails in practice.
Practise Relations and Equivalence
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