Unformatted text preview: 2.2 In general, prove that something is an equivalence relation by proving that it is re f exive, symmetric, and transitive. (a) This is an equivalence that effectively splits the integers into odd and even sets. It is re f exive ( x + x is even for any integer x ), symmetric (since x + y = y + x ) and transitive (since you are always adding two odd or even numbers for any satisfactory a , b , and c ). (b) This is not an equivalence. To begin with, it is not re f exive for any integer. (c) This is an equivalence that divides the non-zero rational numbers into positive and negative. It is re f exive since x ˙ x > . It is symmetric since x ˙ y = y ˙ x . It is transitive since any two members of the given class satisfy the relationship. 5...
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- Fall '08
- Negative and non-negative numbers, Rational number, equivalence class, non-zero rational numbers