We will show that the reflexive closure of transitive closure of a relation is equal to the transitive closure of its reflexive closure through a series of simpler results:
(1) let r1 be a transitive relation on a set s, show that its reflexive closure rr1 is also transitive.
(2) let r2 be a reflexive relation on a set s, show that its transitive closure tr2 is also symmetric.
(3) using the previous results or otherwise, show that r(tr) = t(rr) for any relation r on a set. hint: you may fine the fact that transitive (resp. reflexive) closures of r are the smallest transitive (resp. reflexive) relation containing r useful.

Which question is not a good survey question? a.don't you agree that the financial crisis is essentially over? 63on average, how many hours do you sleep per day? c. what is your opinion of educational funding this year? d.are you happy with the availability of electronic products in your state?

For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius and height 2r minus the volume of two cones, each with a radius and height of r. a cross section of the sphere is and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is the volume of the cylinder with radius r and height 2r is and the volume of each cone with radius r and height r is 1/3 pie r^3. so the volume of the cylinder minus the two cones is therefore, the volume of the cylinder is 4/3pie r^3 by cavalieri's principle. (fill in options are: r/2- r- 2r- an annulus- a circle -1/3pier^3- 2/3pier^3- 4/3pier^3- 5/3pier^3- 2pier^3- 4pier^3)

An airplane is flying at an elevation of 5150 ft, directly above a straight highway. two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is 35 ̊ and to the other is 52 ̊. how far apart are the cars?