Theory of Statistics - George Mason University

0.0 Some Basic Mathematical Concepts 623
Example 0.0.3
Now, again in IR, consider the sequence of intervals
⎧
1
⎨ n
An =
⎩
,
3 1
4 − n for n = 1, 3, 5, . ..
1 1 1
4 − n , 1 + n for n = 2, 4, 6, . ..
In this case,
and
lim supAn
=]0, 1],
n
lim inf
n An
1 3
= , .
4 4
0.0.3 Binary Operations and Algebraic Structures
In a given set S we may find it useful to define a binary operation; that is, a
way of combining two elements of the set to form a single entity. If x, y ∈ S,
we may denote the binary operation as ◦ and we denote the result of the
combination of x and y under ◦ as x ◦ y.
Given a set S and a binary operation ◦ defined on it, various properties of
the operation may be of interest:
closure We say S is closed wrt ◦ iff
x, y ∈ S =⇒ x ◦ y ∈ S.
commutativity We say ◦ defined in S is commutative iff
x, y ∈ S =⇒ x ◦ y = y ◦ x.
associativity We say ◦ defined in S is associative iff
0.0.3.1 Groups
x, y, z ∈ S =⇒ x ◦ (y ◦ z) = (x ◦ y) ◦ z.
One of the most useful algebraic structures is a group, which is a set and an
operation (S, ◦) with special properties.
Definition 0.0.2 (group)
Let S be a nonempty set and let ◦ be a binary operation. The structure (S, ◦)
is called a group if the following conditions hold.
• x1, x2 ∈ S ⇒ x1 ◦ x2 ∈ S (closure);
• ∃ e ∈ S ∋ ∀x ∈ S, e ◦ x = x (identity);
Theory of Statistics c○2000–2013 James E. Gentle