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【Basic Abstract Algebra】Exercises for Section 1.3 — Equivalence relation and equivalence classes

  1. Define a relation R on R2 by stating that (a,b)(c,d) if and only if a2+b2c2+d2. Show that is reflexive and transitive, but itis not symmetric.
    Solution: (1) Obviously, (a,b)(a,b), so is reflexive.
    (2) If (a,b)(c,d), (c,d)(e,f), then we have a2+b2c2+d2, c2+d2e2+f2. So a2+b2e2+f2. Thus (a,b)(e,f), so is transitive.
    (3) Suppose (a,b), (c,d)R2 and a2+b2<c2+d2. Thus (a,b)(c,d). Since c2+d2>a2+b2, (c,d)\not\sim(a,b). Therefore, \sim is not symmetric. #
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