Explorations and Activities 7. Closure Explorations. In Section 1.1, we studied some of the closure prop- erties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: BY NO SA • A set A of numbers is closed under addition provided that whenever x and y are are in the set A, x + y is in the set A. 2.4. Quantifiers and Negations A set A of numbers is closed under multiplication provided that when- ever x and y are are in the set A, x - y is in the set A. • A set A of numbers is closed under subtraction provided that when- ever x and y are are in the set A, x - y is in the set A. For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural num- bers 63 (b) The set of all even integers (c) A = {1, 4, 7, 10, 13,...} (d) B = {..., -6, -3, 0, 3, 6, 9,...} (e) C = {3n+ 1\n € Z} = { ==|₁€N} (f) D =

Problem. Let R be the region given by 0 < z < x² + y² , x² + y? < 3. Evaluate (x + 2)dV. Explanation. This is the region under a paraboloid and inside a cylinder. The reason cylindrical coordinates would be a good coordinate system to pick is that the condition x² + y? < 3 means we will probably go to polar later anyway, so we can just go there now with cylindrical coordinates. The paraboloid's equation in cylindrical coordinates (i.e. in terms of r, 0, and z) is ? z = x + y? Thus, our bounds for z will be くzく Now that we have z, we can look at the cy-plane for our polar bounds. The disc x² + y² < 3 in polar is