Lecture 1: Set Theory
1 Set Theory
One of the main objectives of a statistician is to draw conclusions about a population of objects by conducting an experiment. The fist step in this endeavor is to identify the possible outcomes or, in statistical terminology, the sample space.
Definition 1.1 The set, S, of all possible outcomes of a particular experiment is called the sample space for the experiment.
If the experiment consists of tossing a coin, the sample space contains two outcomes, heads and tails; thus, S = {H, T }.
Consider an experiment where the observation is reaction time to a certain stimulus. Here, the sample space would consists of all positive numbers, that is, S = (0, ∞).
The sample space can be classified into two type: countable and uncountable. If the elements of a sample space can be put into 1–1 correspondence with a subset of integers, the sample space is countable. Otherwise, it is uncountable.
Definition 1.2 An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself).
Let A be an event, a subset of S. We say the event A occurs if the outcome of the experiment is in the set A.
We first define two relationships of sets, which allows us to order and equate sets:
A ⊂ B ⇔ x ∈ A ⇒ x ∈ B (containment) A = B ⇔ A ⊂ B and B ⊂ A. (equality)
Given any two events (or sets) A and B, we have the following elementary set operations:
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Union: The union of A and B, written A ∪ B, is the set of elements that belong to either A or B or both:
A ∪ B = {x : x ∈ A or x ∈ B}.
Intersection: The intersection of A and B, written A ∩ B, is the set of elements that belong to both A and B:
A ∩ B = {x : x ∈ A and x ∈ B}.
Complementation: The complement of A, written Ac, is the set of all elements that are not in A:
Ac= {x : x /∈ A}.
Example 1.1 (Event operations) Consider the experiment of selecting a card at random from a standard deck and noting its suit: clubs (C), diamond (D), hearts (H), or spades (S). The sample space is
S = {C, D, H, S}, and some possible events are
A = {C, D}, and B = {D, H, S}.
From these events we can form
A ∪ B = {C, D, H, S}, A ∩ B = {D}, and Ac= {H, S}.
Furthermore, notice that A ∪ B = S and (A ∪ B)c = ∅, where denotes the empty set (the set consisting of no elements).
Theorem 1.1 For any three events, A, B, and C, defined on a sample space S, a. Commutativity. A ∪ B = B ∪ A, A ∩ B = B ∩ A.
b. Associativity. A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C.
c. Distributive Laws. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
d. DeMorgan’s Laws. (A ∪ B)c= Ac∩ Bc, (A ∩ B)c= Ac∪ Bc.
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The operations of union and interaction can be extended to infinite collections of sets as well.
If A1, A2, A3, . . . is a collection of sets, all defined on a sample space S, then
∪∞i=1Ai= {x ∈ S : x ∈ Ai for some i}.
∩∞i=1Ai = {x ∈ S : x ∈ Ai for all i}.
For example, let S = (0, 1] and define Ai= [(1/i), 1]. Then
∪∞i=1Ai= ∪∞i=1[(1/i), 1] = (0, 1]
∩∞i=1Ai= ∩∞i=1[(1/i), 1] = {1}.
It is also possible to define unions and intersections over uncountable collections of sets. If Γ is an index set (a set of elements to be used as indices), then
∪a∈ΓAa= {x ∈ S : x ∈ Aa for some a},
∩a∈ΓAa= {x ∈ S : x ∈ Aa for all a}.
Definition 1.3 Two events A and B are disjoint (or mutually exclusive) if A ∩ B = ∅. The events A1, A2, . . . are pairwise disjoint (or mutually exclusive) if Ai∩ Aj = ∅ for all i 6= j.
Definition 1.4 If A1, A2, . . . are pairwise disjoint and ∪∞i=1Ai = S, then the collection A1, A2, . . . forms a partition of S.
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