A condition for one domain to be included in another domain?

Question

Consider two domains:

domain1 = Rectangle[{-2.5, -2.5}, {2.5, 2.5}];
domain2[R_] := Disk[{0, 0}, R]

I would like to check whether the second domain is completely included in the first domain. My attempt is

condition[R_] := If[domain2[R] \[Element] domain1, 1, 0]
condition[1]

However, it does not work, as the commands used are incorrect. Could you please tell me how to do this?

Answer 1

domain1 = Rectangle[{-2.5, -2.5}, {2.5, 2.5}];
domain2[R_] := Disk[{0, 0}, R]

You can use RegionWithin:

ClearAll[condition]
condition[R_] := Boole @ RegionWithin[domain1, domain2[R]]

{condition[1], condition[3]}

{1, 0}

You can also use a combination of ForAll and Resolve:

ClearAll[conditionB]
conditionB[R_] := Resolve[ForAll[x, Implies[x ∈ domain2[R], x ∈ domain1]], Reals]

{conditionB[1], conditionB[2], conditionB[3]}

 {True, True, False}

ClearAll[conditionC]
conditionC[R_] := Resolve[ForAll[x, ! x ∈ domain2[R] || x ∈ domain1], Reals]

{conditionC[1], conditionC[2], conditionC[3]}

{True, True, False}

Answer 2

domain1 = Rectangle[{-2.5, -2.5}, {2.5, 2.5}];
domain2[R_] := Disk[{0, 0}, R]
Table[RegionEqual[RegionIntersection[domain1, domain2[R]], 
  domain2[R]], {R, {1, 2, 3}}]

{True, True, False}