Set Theory Basics

What topics are covered in this presentation about basic probability?
The topics covered in this presentation about basic probability are:

  • Set theory.
  • Elements of probability.
  • Conditional probability.
  • Sequential calculation of probability.
  • Total probability and Bayes Rule.
  • Independence.
  • Counting.

What is a set?
A set is a collection of objects.

What are the objects of a set called?
The objects of a set are called elements.

How do you say an element is in a set ?
To say an element is in a set , you write:

What is a set with no elements called?
A set with no elements is called the empty set.

What symbol represents the empty set?
The symbol which represents the empty set is .

What are the three different types of sets?
The three different types of sets are:

  1. Finite -
  2. Countably infinite -
  3. Uncountable - A set which takes a continuous set of values like the interval , the real line, etc.

What is the other notation which can be used to describe a set?
The other notation which can be used to describe a set is set builder notation.

Example of set builder notation for describing a set containing the interval

What does mean if both and are sets?
If and are sets, means that all of the elements in the set are in the set .

What is a set called if all of the elements in it are also in a set ?
If all of the elements in a set are also in a set , then B is called a subset of set .

What is the universal set?
The universal set is a set containing all of the elements under discussion, like the sample space for a random experiment.

What symbol represents the universal set?
The symbol which represents the universal set is .

What is complementation?
Complementation is creating the complement of a set with respect to another set.

What is the complement of a set with respect to a set ?
The complement of a set is the set of all elements which aren't in set but exist within set .

How do you describe the complement of a set with respect to a set in set builder notation?
To describe the complement of a set with respect to a set in set builder notation, you write:

What is the complement of the universal set?
The complement of the universal set is the empty set.

What is an intersection?
An intersection is the set of all elements which are in both sets.

How do you describe the intersection of a set and a set in set builder notation?
To describe the intersection of a set and a set in set builder notation, you write:

What is a union?
A union is the set of all elements that are in either set.

How do you describe the union of a set and a set in set builder notation?
To describe the union of a set and a set in set builder notation, you write:

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What does it mean for a set and a set to be disjoint or mutually exclusive?
A set and a set are disjoint or mutually exclusive when they don't have any elements in common.

What is the intersection of disjoint sets equal to and how is this expressed?
The intersection of disjoint set is equal to the empty set and it's expressed as:

What does it mean for a collection of sets to partition a set and how is this expressed?
A collection of sets partitions a set when they're disjoint and the union of all of them equals set , and it's expressed as:

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Examples of Venn Diagram representations for sets, complements, intersections, and unions

Venn Diagram Representations For Sets, Complements, Intersections, and Unions.png

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