Exercise 1-1

Give an example and non-example for

  1. an order-preserving function

  2. a metric-preserving function

  3. an addition-preserving function

Solution(1)

  1. Order-preserving \(x+1\), non-order-preserving \(-x\)

  2. Metric preserving \(x+1\), non-metric-preserving \(2x\)

  3. Addition-preserving \(2x\), non-addition-preserving \(x^2\)