Raising the bar

Chapter 4. Probability

Unit 4.1. Set operations

On the surface it is simple: the chances are 50:50 that on the coin we get either heads or tails. Don't be fooled by this simplicity. The way uncertainty is modeled in the theory of probabilities is complex: from sets to probabilities to random variables and their characteristics.

Recall that the notion of a set is not defined. We just use equivalent names (collection, family, or array) and hope that, with practice, the right intuition will develop.

In the definitions below $latex S$ denotes the set of all students of a particular university, $latex A$ denotes the set of students taking an anthropology class and $latex B$ is the set of students taking a biology class. We write $latex x \in A$ to mean that $latex x$ is an element of $latex A$ ($latex x$ belongs to $latex A$). $latex x \notin A$ means that $latex x$ is not an element of $latex A$. Read the definitions and avoid a common mistake: in the union of $latex A$ and $latex B$, the elements of $latex A$ and $latex B$ are included only once. For example, if $latex A = {1,\,2},\ B = {2,3}$ then $latex A \cup B = {1,\,2,\,3}$ and writing $latex A \cup B = {1,\,2,\,2,\,3}$ would be wrong.

Set operations: definitions, visualization and logic.

	Definition 4.1. The union $latex A \cup B$ is the set of elements which belong to $latex A$, to $latex B$ or both. Comment. Logically, the union corresponds to nonexclusive "or". $latex A \cup B$ means all students who study anthropology or biology (or both).
	Definition 4.2.  The intersection $latex A \cap B$ is the set of elements common to $latex A$ and $latex B$. Comment. Logically, the intersection corresponds to "and". $latex A \cap B$ is the set of students who study both anthropology and biology.
	Definition 4.3. The difference $latex A \backslash B$ is the set of elements of $latex A$ which do not belong to $latex B$. Comment. $latex A \backslash B$ is the set of students who study anthropology but not biology. Don't use "$latex -$" (minus) or "$latex |$" (pipe) instead of the backslash "$latex \backslash$".
	Definition 4.4. We say that $latex A$ is a subset of $latex S$ and write $latex A \subset S$ if all elements of $latex A$ belong to $latex S$. Comment. In applications $latex S$ will be fixed, while its subsets will be changing.
	Definition 4.5. A complement (or complementary event) of $latex A$ is defined as the difference $latex \bar A = S \backslash A$ and includes elements of $latex S$ outside $latex A$. Comment. $latex \bar A$ is the set of students who do not study anthropology.
	Definition 4.6. For convenience, people introduce the empty set $latex \varnothing$ (the set without elements). The empty set plays the role of zero: for any set $latex A$, one has $latex A = A \cup \varnothing$.
	Definition 4.7. $latex A$ and $latex B$ are called disjoint (or nonoverlapping) if $latex A \cap B = \varnothing$ (they have no common elements).

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