# Simplify Expressions used to define Region

For example,

RegionPlot[{x + y > 1, x - y > 1, y - x > 1, -x - y > 1}, {x, -2, 2}, {y, -2, 2}]


can be somehow written as:

RegionPlot[{Abs[x] + Abs[y] > 1}, {x, -2, 2}, {y, -2, 2}]


My question is whether Mathematica can simplified the first group of expressions to the second one?

I think I do not make the question clear. I think conditions in

{x + y > 1, x - y > 1, y - x > 1, -x - y > 1}

describing the same region as

{Abs[x] + Abs[y] > 1}

Therefore, the question is whether {x + y > 1, x - y > 1, y - x > 1, -x - y > 1} can be reduce to {Abs[x] + Abs[y] > 1}. or, generally, is there any method can simplify the expressions used to describe a region?

RegionPlot[
Or @@ {x + y > 1, x - y > 1, y - x > 1, -x - y > 1}, {x, -2,
2}, {y, -2, 2}]


suffices

• Sorry, I think I do not ask a clear question. If you have time, please check the updated question. Thanks! – Kattern Apr 20 '16 at 13:55
• @Kattern I apologize for delay. Timezone differences...I hope your aim has been achieved:) – ubpdqn Apr 21 '16 at 4:43
• No, maybe it is not a good idea to try to simplify those expressions. – Kattern Apr 21 '16 at 6:20

I haven't been able to turn one into the other, but they can both be reduced to the most basic form, which is the same:

exp1 =
Reduce[x + y > 1 || x - y > 1 || y - x > 1 || -x - y > 1, {x, y},
Reals]
exp2 = Reduce[Abs[x] + Abs[y] > 1, {x, y}, Reals]
exp1 == exp2
(* x < -1 || (-1 <= x <= 0 && (y < -1 - x || y > 1 + x)) || (0 <
x <= 1 && (y < -1 + x || y > 1 - x)) || x > 1 *)
(* x < -1 || (-1 <= x <= 0 && (y < -1 - x || y > 1 + x)) || (0 <
x <= 1 && (y < -1 + x || y > 1 - x)) || x > 1 *)
(* True *)


and it can be plotted the same,

RegionPlot[exp1, {x, -2, 2}, {y, -2, 2}]


giving the same output as in ubpdqn's answer