Let I notate the interval [0,infinfty). For each r ( that\'s an element of) I de

Question

Let I notate the interval [0,infinfty). For each r (that's an element of) I define,

          Ar = {(x,y) (that's an element of) R x R : x2 + y2 = r2},

          Br = {(x,y) (that's an element of) R x R : x2 + y2 (less than or equal to) r2},

          Cr = {(x,y) (that's an element of) R x R : x2 + y2 > r2}.

(a) Determine Ur (that's an element of) I Ar and (intersection of) r(that's an element of) I Ar.

(b) Determine Ur (that's an element of) I Br and (intersection of) r(that's an element of) I Br.

Explanation / Answer

The sets Ar and Br are exactly the same... I think one of them should have a "less than" sign. I assume those questions marks I see should be the symbol for "is an element of." And I also assume that x2 and y2 mean x2 and y2 and so on, and I assume "U" is a union and "n" is an intersection. The set A or B, whichever one with equality, will be the set of points that are on a circle of radius r. The set with the less-than will be the set of points inside a circle of radius r. The set Cr will be the set of points outside a circle of radius r. I will assume A is the equality set (set of point on the circumference of a circle) and I assume B will be the less-than set (set of points inside the circle.) a) The union of all the points on the circumference of circles of radius 0 to 8 (not including 8) is {(x,y) : x2 + y2 < 82} The intersection is the empty set since no two circumferences intersect. b) The union of all the points in the interiors of the circles will be the largest circle, so the union is {(x,y) : x2 + y2 < 82}. The intersection is the smallest set (i.e. when r=0) so the intersection is {(0,0)} c) The union of all the points in the exterior of the circles is the biggest "outside" so this happens when r=0. The union is { RxR (0,0) } (all the points in the space except the origin.) The intersection of all these sets is {(x,y) : x2 + y2 > 82}

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