Using logical OR to combine inequalities.I have a physical system that must satisfy one of two inequalities: x ≤ y or p ≤ qBut not necessarily both simultaneously.Is there a way to combine this into a single inequality?

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Using logical OR to combine inequalities.I have a physical system that must satisfy one of two inequalities:  x  ≤  y or   p  ≤  qBut not necessarily both simultaneously.Is there a way to combine this into a single inequality?

ejigaboo8y · Accepted Answer

Let   a  =  y  −  x and   b  =  q  −  p. We want   a  ≥  0 OR   b  ≥  0. Equivalently, we want   max  {  a  ,  b  }  ≥  0.We have   max  {  a  ,  b  }  =            a      +      b        2    +                |      a      −      b          |        2  . (This is a standard expression for the max of two numbers--geometrically the midpoint of the two numbers, plus half the distance between them.)So for the inequality to hold, we want             a      +      b        2    +            ∣      a      −      b      ∣        2    ≥  0..We can now put this in terms of the original variables:  x  ≤  y  OR   p  ≤  q is equivalent to             (      y      −      x      +      q      −      p      )      +      ∣      y      −      x      −      q      +      p      ∣        2    ≥  0

Using logical OR to combine inequalities. I have a physical system

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