What is the interior of S? b. What is the boundary of S? c. Is S open, closed, or neither? All of the notions introduced in this lecture can be applied to sets of real numbers: write x for x and |x| for k x k. A neighborhood of a number

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.

2. Define the absolute value of x E R. then find all real numbers x such that |x +1 < x – 1).

Select all the real numbers M among the choices below such that the two planes given by 1 = 1 r. and r. 2 M do not intersect. [Note: In this question, you can select more than one correct answer] O a. Mequals -13 O b. M equals 2 O c. M equals 1