【Basic Abstract Algebra】Exercises for Section 1.3

2. Define a relation \(R\) on \(\mathbb R^2\) by stating that \((a,b)\sim(c,d)\) if and only if \(a^2+b^2\le c^2+d^2\). Show that \(\sim\) is reflexive and transitive, but itis not symmetric.
   Solution: (1) Obviously, \((a,b)\sim(a,b)\), so \(\sim\) is reflexive.
   (2) If \((a,b)\sim(c,d),~(c,d)\sim(e,f)\), then we have \(a^2+b^2\le c^2+d^2,~c^2+d^2\le e^2+f^2\). So \(a^2+b^2\le e^2+f^2\). Thus \((a,b)\sim(e,f)\), so \(\sim\) is transitive.
   (3) Suppose \((a,b),~(c,d)\in \mathbb R^2\) and \(a^2+b^2< c^2+d^2\). Thus \((a,b)\sim (c,d)\). Since \(c^2+d^2>a^2+b^2\), \((c,d)\not\sim(a,b)\). Therefore, \(\sim\) is not symmetric. #