Set theory
1. Let X be the set of 2 × 2 matrices. Define
d : X × X → R; d(x, y) = max
1≤i,j≤2
{|xi,j − yi,j |}
where xi,j is the i, j-th entry of x. Prove or disprove that this is a metric on X.
2. Let A =
2
n
: n ∈ Z
. Does A have a supremum and/or an infimum? If so, prove it.
3. Let A =
2
n
: n ∈ Z
. Determine if A is a closed set. Prove or disprove from the
definition of a closed set.
4. Let X be a set. Define
d : X × X → {0, 1} ⊆ R
d(x, y) =
0 x = y
1 x 6= y
.
Let x0 ∈ X, consider the open ball of radius 1 around x0,
B(x0, 1) = {x ∈ X : d(x, x0) < 1}
the closed ball of radius 1 around x0,
B¯(x0, 1) = {x ∈ X : d(x, x0) ≤ 1}
and the closure of the open ball of radius 1 around x0, denoted B(x0, 1). Determine,
explicit descriptions of these sets and prove your answers.
DETAILED ASSIGNMENT
