Subset of a set

This lesson will explain what a subset of a set is. We will start with a definition


Definition:

Set B is a subset of a set A if and only if every object of B is also an object of A.

We write B ⊆ A

By definition, the empty set( { } or ∅ ) is a subset of every set

Now, take a look at the following Venn diagrams.

Definition of Venn Diagrams:

Venn Diagrams are closed circles, named after English logician Robert Venn, used to represent relationships between sets


Venn diagrams


B = { a, b, c}

A = { a, b, c, f}

U = { a, b, c, f}

Since all elements of B belong to A, B is a subset of A

Proper subset:

Set B is a proper subset of set A, if there exists an element in A that does not belong to B.

we write B ⊂ A

Having said that, B is a proper subset of A because f is in A, but not in B.

We write B ⊂ A instead of B ⊆ A

Universal set:

The set that contains all elements being discussed

In our example, U, made with a big rectangle, is the universal set

Set A is a proper subset of U because not all elements of U are in subset A

Notice that B can still be a subset of A even if the circle used to represent set B was not inside the circle used to represent A. This is illustrated below:

Venn diagram


As you can see, B is still a subset of A because all its objects or elements (c, and d) are also objects or elements of A.

B is again a proper subset of A because there are elements of A that does not belong to B

A and B are also subsets of the universal set U, but especially proper subsets since there are elements in U that does not belong to A and B

In general, it is better to represent the figure above as show below to avoid being redundant.

Venn diagram


The area where elements c, and d are located is the intersection of A and B. More on this on a different lesson!

If you have any questions about this lesson, I will be more than happy to answer them.


Subset of a set quiz. Use the quiz below to see how well you can recognize subsets.
 




Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest


Math quizzes

 Recommended

Scientific Notation Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz

Types of angles quiz


Tough algebra word problems

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

K-12 math tests


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.