Why is the Empty Set a Subset of Every Set? | Set Theory, Subsets, Subset Definition

The empty set is a very cool and important part of set theory in mathematics. The empty set contains no elements and is denoted { } or with the empty set symbol ∅. As a result of the empty set having no elements is that it is a subset of every set. But why is that? We go over that in this video!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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If you are preparing for Set Theory or in the midst of learning Set Theory, you might be interested in the book I learned set theory and proofs from. It is “Book of Proof“ by Richard Hammack. Check out the book and see if it suits your needs! You can purchase the textbook using the affiliate link below which costs you nothing extra and helps support Wrath of Math!

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[brysene]

I see so the Empty Set is a subset of every other set because of a technicality. Thank you for your explanation .

[divyanshbhatt]

i have a question, why did you put equal under the subset sign early in the video

[suwilanjisimbule]

Thank you so much you explained it perfectly

[ejsafara]

oh my god, such amazing explanation, now i understand it, thousand thanks! :D

[The-Uncertainty]

Well explained brother. It's clear for me, finally.

[rishabhnarula]

I think the empty set being a subset of every set is like a notion of the idea that if everything in the universe is seen just as number of things, then the number representing collection of smallest number of things would be there in the number representing any collection of all things.

or another way of looking at it is if any collection of things is considered as a universe itself, then of course you could keep taking things away from it to make the collection smaller, and conclude this collection of things was inside what you originally had and conclude that nothing must be in it by the end of the process.

hence nothing is in everything.

One more idea is if it is not a subset of some set, then there must be something in it that is not there in the other set.

there is nothing in it.

hence the contradiction.

[weldonwebb]

I concur and I compliment you for your use of Proof. Since on the definition of what is a subset, this definition does not apply to sets with no elements. So {} is a subset of of sets including the universal set

[subinssubins]

How many subset do a null set have and is set R=Set U

[curiosity]

Hi! Thank you for your explanation. I had a question: is the empty set a proper subset of every other non-empty set?
If A = {1, 2, 3, 4} and B = {1, 3}, B is a proper subset of A because it contains elements of A but is not equal to it. So if we're asked to write all the proper subsets of A, do we also include the empty set or not? Thank you!

[josnaakter]

If a set needs to be a subset of another set it needs to contain at least one element that is also an element of another set. But when we compare a set that has elements in it and empty set, say, A={1, 2, 3}
And empty set, then if i say that empty set is a subset of set A because it contains no elements that are disqualifing it from being a subset of A then we also need to show, that one element that is also present in the set A. So how can we say that empty set is a subset of any set?

[siulapwa]

Sounds weird coz if you think of a subset A of B to mean every element of A must also be in B, then since the empty set has no elements, it seems there should be no overlap between the empty set and any nonempty set, they have nothing in common, (no pun intended) the only way it makes sense is if you define sets to made of empty which I'm not sure is true for all sets

[rainily]

Is the empty set also a subset of a cartesian product A x B?

[justinswag]

Very logical proof, thank you I finally understand this SHIT !

[stephaniepaguinto]

This a very great explanation. But I do have one question, can a set be a subset of an empty set? If an empty set is subset of every set, then what about the other way?

[princesskurtneycalitis]

but will it be a proper or improper subset??

[gloriakalengelayi]

Hi, an equivalent set can be a subset of its equivalence. All subsets are subsets of themselves because of the equal rule. Now a proper subset has the same elements with the exception of at least 1 element. So the empty set meets the criteria of having at least 1 element less but it does not meet the criteria of have the same element because nothing is not the same as something. In fact they are the opposite. The empty set being a part of every set does not meet all the rules to either fit into the equivalent, equal or proper subset.

[gokulkrish]

What is the difference between subset and proper subset and what does improper set mean ?

[mollymorua]

Taking college algebra for the second time. Watching this I feel like the biggest idiot on planet Earth because you made this so easy...

[takeda]

Thanks. but I need more explanation cuz i needed this for my homework but thanks! (I was confused when my teacher explained this)

[haileyy]

Shouldn't it be a strict subset since the elements in the empty set are not equal to the elements of set A?