Probability Theory: Lecture Note 2 - Set and Set Operations part I

Introduction:

A set is a collection of objects.

The objects in a set are called elements of the set.

A well–defined set is a set in which we know for sure if an element belongs to that set.

Example:

– The set of all movies in which Leonardo DiCaprio appears is well–defined.

–The set of all movie serials made by ZEE TV is well–defined.

–The set of best TV shows of all time is not well–defined. (It is a matter of opinion.)

Notation:

When talking about a set we usually denote the set with a capital letter.

Roster notation is the method of describing a set by listing each element of the set.

Example: Let C = The set of all movies in which John Cazale appears. The Roster notation would be C={The Godfather, The Conversation, The Godfather II, Dog Day Afternoon, The Deer Hunter }. (All 5 of these movies were nominated for Best Picture by the Motion Picture Academy.)

Example: Let set A = The set of odd numbers greater than zero, and less than 10. The roster notation of A={1, 3, 5, 7, 9}

Sometimes we can’t list all the elements of a set. For instance, Z = The set of integer numbers. We can’t write out all the integers, there infinitely many integers. So we adopt a convention using dots …

The dots mean to continue on in this pattern forever and ever.

Z = { …-3, -2, -1, 0, 1, 2, 3, …}

W = {0, 1, 2, 3, …} = This is the set of whole numbers.

Set – Builder Notation:

When it is not convenient to list all the elements of a set, we use a notation the employs the rules in which an element is a member of the set. This is called set-builder notation.

A = {x | x > 5} = This is the set A that has all real numbers greater than 5.

The symbol | is read as such that.

Universal Set, Superset, and Subsets:

•The Universal Set denoted by U is the set of all possible elements used in a problem.

•When every element of one set is also an element of another set, we say the first set is a subset and the second set is a superset.

•Example A={1, 2, 3, 4, 5} and B={2, 3},

We say that B is a subset of A. A is the superset.

The notation we used,