Subsets
Hi everyone. This math post is an introduction to subsets from the mathematical field of set theory.
Introduction
The basic idea of a subset is having a smaller group being part of a larger group. Examples include:
- Students belonging to a school.
- A video game from a collection of video games.
- The produce (fruits and vegetables) section in a grocery store/supermarket.
- A slice of cake.
- A few candies from the candy jar.
- A few countries from a continent.
The Definition Of A Subset
From the field of mathematics, a subset refers to a set or a collection of objects/elements that is a part of a larger set/collection.
If set B has the same elements as set A but set B has less elements then B is a proper subset of set A. This is denoted by B ⊂ A or A ⊃ B.
Example One
Set A = {1, 2, 3, 4, 5}$ is a (proper) subset of the set B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} since set A contains the same elements (the numbers 1, 2, 3, 4 and 5) as set B. Also, set A is a smaller set than set B. In math notation, we denote this as A ⊂ B.
Example Two
Suppose you have a local grocery store and a larger supermarket. Denote the set of items for the grocery store as L and denote the set of items for the larger supermarket as M. The two stores have the following items:
The set L is a proper subset of M. (L ⊂ M)
Example Three
If the local grocery store had an item such as cherries along with apples, oranges, noodles, potatoes and onions then the set L is not a subset of M since the larger supermarket does not carry cherries.
Example Four - Empty Set Case
The empty set denoted by ∅ represents nothing. The empty set is a subset of any set and is a proper subset of any non-empty set. For example, the empty set ∅ is a subset of A = {a, b, c} denoted by ∅ ⊂ A.
The Number Of Subsets In A Set (Example)
Given a set, you can determine the number of subsets in that set. Consider the set A = {a, b, c} again.
Subsets of the set A include:
- The empty set (∅)
- The set A itself which is {a, b, c}
- {a}
- {b}
- {c}
- {a, b}
- {b, c}
- {a, c}
The set of the subsets is called the Power Set of A. In this example, the Power Set is given by:
Extension - Sigma Algebras
One extension of set theory is the concept of sigma algebras. Although I learned a little bit of sigma algebras in a probability course (stochastic calculus), my memory of it is not very good.
EDIT (Sept. 4 10:53AM EST): Realized there was a subtle but big mistake in my second example. It is fixed.