1. Quizz 2
Name: ( )
Student ID number:( )
Instruction:
(1) When you say that a set is nonempty, you need to find an element belonging to the set.
(2) When you say that a set A is a subset of another set B, you need to show that elements of A belong to B.
(3) When you say that two sets A, B are equal A = B, you need to show that A ⊂ B and B ⊂ A.
(4) When you say that a point p is an interior point of a set A ⊂ Rn, you need to find the radius explicitly so that B(p, ) is contained in A.
(5) When you say that p ∈ A, you need to show that B(p, ) ∩ A is nonempty for every
> 0.
(6) When you say that p ∈ ∂A, you need to show that B(p, ) ∩ A and B(p, ) ∩ (Rn\ A) are both nonempty for every > 0.
(7) When you say that p is an isolated point of A, you need to find > 0 so that B(p, ) ∩ A = {p}.
True or false.
(1) ( ) The origin (0, 0) is an accumulation point of U = {(x, y) ∈ R2 : x > 0}.
(2) Let A = {(m, 0) ∈ R2 : m ∈ Z}.
(a) ( ) The set int(A) is nonempty.
(b) ( ) Let A0 be the set of all accumulation points of A. Then A0 = ∅.
(c) ( ) A consists of isolated points only.
(d) ( ) A is a closed subset of R2.
(3) Let D = {x2+ y2 < 1} and p = (0, 1) and O = (0, 0).
(a) ( ) p is an adherent point of D.
(b) ( ) p is an exterior point of D.
(c) ( ) p is an isolated point of D.
(d) ( ) p is a boundary point of D.
(e) ( ) O is an interior point of D.
(f) ( ) O is an adherent point of D.
(g) ( ) O is an accumulation point of D.
(h) ( ) The boundary of D is {x2+ y2 = 1}.
(4) Let S = {(x, y) ∈ R2 : 0 ≤ x ≤ 1, 3 ≤ x ≤ 5}.
(a) ( ) int(S) = {(x, y) ∈ R2 : 0 < x < 1, 3 < x < 5}.
(b) ( ) ∂S = {(x, y) ∈ R2: x = 0, x = 1, x = 3, x = 5}.
(5) ( ) An isolated point of a subset of Rn belongs to the set.
(6) ( ) An isolated point of a subset of Rn can be an interior point of that set.
(7) ( ) An accumulation point of a subset of Rn could be the interior point of that set.
(8) ( ) An accumulation point of a subset of Rnmust be the interior point of that set.
(9) ( ) An accumulation point of a subset of Rn can never be the exterior point of that set.
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