# Algebra 2 – Set Equality and Subsets

Hello. I’m Professor Von Schmohawk and welcome to Why U. In the previous lecture
we saw how to define sets and their members through the use of set-builder notation. In this lecture, we will learn about several
types of relations between sets. One such relation is “equality”. If two sets contain exactly the same elements
they are said to be equal. For example, set A, which consists of
the elements one, two and three and set B consisting of
the elements two, three, and one are equal since they both contain
the same elements. Notice that it doesn’t matter that their elements
are listed in different orders. This is because the order in which the elements
of a set are listed is irrelevant. The set consisting of one, two, and three is exactly the same as the set
consisting of two, three, and one. The list does not imply that there is any
particular order to the elements. It only says that every element listed
is a member of the set. So if sets A and B contain the same elements
then they are equal. Another important relation between sets
is the relation of “subset”. If all the elements of set A
are also contained in set B then we say that set A
is a subset of set B. This relation is denoted using
the subset symbol. If two sets, A and B, are equal
then they are subsets of each other. This is because all the elements of A
are members of B and all the elements of B
are also members of A. So any two sets that are equal
are subsets of each other. And since every set is equal to itself
every set is also a subset of itself. So the subset symbol can be used whenever all the elements of A
are also members of B whether A has the same number of elements
as B or has fewer elements. But when A has fewer elements, we can be more
specific and call A a proper subset of B. We denote this using the proper subset symbol which is the subset symbol
without the line underneath. In addition to the relation of subset,
a set can also be a “superset”. In any case where set A
is a subset of set B we can also say that set B
is a superset of set A. This relation is denoted using the superset
symbol, which is the subset symbol reversed. Likewise, if A is a proper subset of B
then B is a proper superset of A. But what about the empty set,
the set containing no elements? The empty set is considered to be a subset
of every other set including itself. In fact, the empty set is the only set
which is a subset of every set. So far we have represented sets
by listing their members or using set-builder notation or by drawing little ovals
with the elements inside. In the next lecture, we will see how to
visualize sets using Venn Diagrams.

### 26 thoughts on “Algebra 2 – Set Equality and Subsets”

• July 28, 2012 at 7:06 am

i like your revolutioninzed way of teaching….

• October 15, 2012 at 11:21 pm

only 2 comments and only 2500, subscribers, he deserves a lot more people. Oh yeah, only 16 likes (I just liked đ

• November 7, 2012 at 11:32 am

Very useful for teaching middle school. Thanks NyWhyU đ

• November 7, 2012 at 1:35 pm

• February 26, 2013 at 11:25 am

hi .. can i download ur video … it so very helpful for student here to learn …

• June 12, 2013 at 10:57 pm

A very nice representation of set, subsets, and supersets! (Discrete Math-VCU).

• September 12, 2013 at 9:12 am

thanks man very innovative way of teaching

• April 22, 2014 at 5:20 pm

woot, math…

• May 9, 2015 at 7:59 pm

great way to explain

• October 13, 2015 at 6:53 pm

on the proper subset, what If A has fewer elements than B, also A has an element that B does NOT. Would this still make A a proper subset of B?

• February 7, 2016 at 9:21 pm

Unbelievable these videos should have way more subscribers and views!

• March 8, 2016 at 12:53 pm

• May 25, 2016 at 6:57 am

03:30 And this makes a problem when we try to count the number of elements in a set: if an empty set is always a subset of every set, that means that it also needs to be counted as an element when we have a set of sets. And this is the case with the set-theoretical definition of numbers, which is based on nesting empty sets one inside another and counting them.

• May 25, 2016 at 7:00 am

+quantumsingularityup not really. In order to reason about sets, you need the rules of reasoning in the first place. Therefore, logic is more basic than set theory (and perhaps the most basic in all Mathematics).

• October 16, 2016 at 6:16 pm

then what's the difference between equality and subset ?

• November 25, 2016 at 11:45 am

That into is like powerpuff girls

• March 5, 2017 at 4:20 pm

Well Explained. Thanks.

However, I'm wondering about the tobacco smoking pipes.
tobacco smoking pipes!? Are you promoting tobacco !?

• March 7, 2017 at 3:25 pm

lol love it i love whyu

• May 16, 2017 at 7:40 pm

Can you make videos in Combinatorics?

• July 25, 2017 at 6:09 am

This is a lot easier and more fun than being shoved into a classroom at 7:00am while people throw rubber bands and insult you across the room while the teacher corrects homework all day instead of teaching and then fail because you forget to turn in your work :).

• October 7, 2017 at 5:25 am

I want to make a set of all fractions.
Can you please tell me how to say that a fraction has to be simplified in notation? (note: the fraction is one variable, not two seperate variables)

• February 2, 2019 at 2:31 pm

That video is fire

• March 25, 2019 at 10:58 am

Why is the empty set considered to be a subset of every set? A set other than an empty set is not empty (has some number of elements), how are they even related?

• May 1, 2019 at 12:19 am

Ummmm the lady has 1,48 foods she ate 1,67 how many DOSE she have leftover heheheheheh MATH ppl