Let X = {-1, 0, 1} and A = P(x) and define a relation R on A as follows: For all sets s and t in P(x), s Rt e the sum of the elements in s equals the sum of the elements in t. It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets Enter EMPTY or Ø for the empty set.)

State the formal definitions that a relation must satisfy in order to count as Reflexive, Symmetric, and Transitive. Then construct three small worlds showing, respectively, that a relation can be Reflexive and not Symmetric, Symmetric and not Transitive, and Reflexive and not Transitive. Finally, for each of the three small worlds that you construct, give an example of a relation in ordinary language that works like the relation in the small world you built. That is, for your first small world, give an ordinary language example of a relation that is Reflexive and not Symmetric.

please answer number 7. If you can please answer as much as you can.

Consider the initial value problem given below. y' = 0.9-y+y³, y(0)=0 Use the improved Euler's method with tolerance to approximate the solution to this initial value problem at x = 1.1. For a tolerance of ε = 0.004, use a stopping procedure based on the absolute error. To express the improved Euler's method with tolerance for the given differential equation, determine the values of xo, Yo, C, N, and f(x,y). Here, M is a safeguard determined by the user. Xo = Set z = Yo = C= C For m=0 to M, N = f(x,y)=