Consider E={-2+1/n} n=1 to infinity U (3,9) as a subset of R with the usual definition of < (less than) a

Q: State whether the following statements are true or false: 1. Let S be a non-empty subset of the set of real numbers R. Ir s is bounded above, then Sups is exist, but need not to be unique in general. 2. If A = (-5,5) and B = (5,10), then Inf(A + B) = 10 and Sup(A + B) = 15. 3. The closed interval [1,2] has no maximal element. 4. The set of natural numbers N of R is unbounded. 5. The set of real numbers R, has Sup(R) = o and Inf(R) = -co. 6 The set S= (x E RỊ x? - 25 s 0} has Max(S) = 5 and Inf(S) = -5 with no minimal %3D element. 7. The set S = {1+nez*} has Max(S) = 2 and Min(S) = 1. 8. Every bounded set of real numbers R has maximal and minimal elements. 9. The properties (M2) and (M2) of the definition of the metric space are state that the distance from any point to another is never negative, and that the distance from a point to itself is zero. 10. There are many metric functions d: M x M -R that can be defined on a non-empty set M.

b) Prove or disprove: i. [x]-[x]. ii. For all positive real numbers x and y, [xy] ≤ [x] - [y].

The floor function is defined as bxc = max{m ∈ Z | m ≤ x}. Effectively, the floor function is the action of truncating the decimals off of a real number.(a) EvaluateZ n0bxc dx ,where n is a positive integer.(b) EvaluateZ babxc dx ,where a and b are real numbers with 0 ≤ a < b