3697 words - 15 pages

1. Preliminaries

1. Set operations

A set is a collection of some items such as outcomes of an experiment.

We denote sets using upper case letters, say A and write a ∈ A if a is an

element belonging to A.

If A and B are two sets, then the notation A ⊆ B means that the set A is

included in B , i.e. each element of A is also an element of B :

A⊆B

iff [∀a ∈ A : a ∈ B ]

If A ⊆ B and A = B then we say that the set A forms a proper subset of B

and write A ⊂ B .

Dorota M. Dabrowska (UCLA)

Biostatistics 255

September 21, 2011

1 / 49

In what follows all sets will be subsets of a larger set Ω. The complement

of A in Ω is denoted by Ac and represents elements of Ω which do ...view middle of the document...

Dabrowska (UCLA)

and

Biostatistics 255

A ∩ (A ∪ B ) = A

September 21, 2011

5 / 49

Lastly, complementation is governed by

(viii) De Morgan laws:

if A ⊆ B

then

(A ∩ B )c = Ac ∪ B c

Dorota M. Dabrowska (UCLA)

Biostatistics 255

B c ⊆ Ac

and

(A ∪ B )c = Ac ∩ B c

September 21, 2011

6 / 49

The operations of union and intersections are also deﬁned for arbitrary

families of sets. Let T be an arbitrary index set, and suppose that At ⊆ Ω

for each t ∈ T . Then

At = {ω ∈ Ω : ∃ t ∈ T ω ∈ At }

t ∈T

At = {ω ∈ Ω : ∀ t ∈ T ω ∈ At }

t ∈T

We have

(i)

t ∈T

At ⊆ At0 ⊆

t ∈T

At

for all

t0 ∈ T

(ii) if At ⊆ Bt for all t ∈ T then

At ⊆

t ∈T

Dorota M. Dabrowska (UCLA)

Bt

At ⊆

and

t ∈T

Biostatistics 255

t ∈T

Bt

t ∈T

September 21, 2011

7 / 49

(ii) De Morgan laws

At ]c =

[

t ∈T

Ac

t

and

t ∈T

[

At ]c =

t ∈T

Ac

t

t ∈T

(iv) associativity laws:

At ∪

t ∈T

[At ∪ Bt ] and

Bt =

t ∈T

Dorota M. Dabrowska (UCLA)

t ∈T

At ∩

t ∈T

Biostatistics 255

[At ∩ Bt ]

Bt =

t ∈T

t ∈T

September 21, 2011

8 / 49

Cartesian products

If Ω and Ω′ are two sets then their Cartesian product Ω × Ω′ represents the

collection of ordered pairs (ω, ω ′ ) such that ω ∈ Ω and ω ′ ∈ Ω′ . More

generally, if A ⊆ Ω and B ⊆ Ω′ then

A × B = {(ω, ω ′ ) : ω ∈ A

Dorota M. Dabrowska (UCLA)

Biostatistics 255

and

ω′ ∈ B}

September 21, 2011

9 / 49

We have

A × B = ∅ iff

A=∅

or B = ∅

(A1 ∩ A2 ) × (B1 ∩ B2 ) = A1 × B1 ∩ A2 × B2

[A × B ]c = Ac × Ω2 ∪ Ω1 × B c

A × (B1 − B2 ) = A × B1 − A × B2

(A1 − A2 ) × B = A1 × B − A2 × B

Dorota M. Dabrowska (UCLA)

Biostatistics 255

September 21, 2011

10 / 49

A×

A × Bt

Bt =

t ∈T

A×

t ∈T

A × Bt

Bt =

t ∈T

t ∈T

At × B =

t ∈T

At × B

t ∈T

At × B =

t ∈T

At × B

t ∈T

Finally

if

A1 ⊆ A2

and

B1 ⊆ B2

then

A1 × B1 ⊆ A2 × B2

Dorota M. Dabrowska (UCLA)

Biostatistics 255

September 21, 2011

11 / 49

2. Extended real line

The extended real line, denoted by R , consists of all real numbers with

added ∞ and −∞.

The addition and multiplication of ﬁnite reals is deﬁned in the usual way.

We also make the following conventions. For any ﬁnite real a

(i )

a+∞=∞+∞=∞

a − ∞ = (−∞) − ∞ = −(∞) + (−∞) = −∞ − (−∞) = ∞

(ii )

if

a > 0 then

a · ∞ = ∞ · ∞ = (−∞) · (−∞) = (−a)(−∞) = ∞

a · (−∞) = (−∞) · ∞ = ∞(−∞) = (−a)∞ = −∞

(iii )

0 · ∞ = 0 = 0 · (−∞)

Dorota M. Dabrowska (UCLA)

Biostatistics 255

September 21, 2011

12 / 49

(iv )

a

a

=

=0

∞

−∞

The remaining operations (such as ∞ − ∞ or

Dorota M. Dabrowska (UCLA)

Biostatistics 255

∞

∞)

are undeﬁned.

September 21, 2011

13 / 49

If A is a subset of reals then the least upper bound of A is denoted by

sup A and is deﬁned as the smallest extended real number a such that

x ≤ a for all x ∈ A. Thus

∀x...

Beat writer's block and start your paper with the best examples