What is a Null Set | Is Null Set a Subset of Every Set? | Don't Memorise

In this video, we will learn:
0:00 what is a null set?
0:40 how is the null set represented?
0:53 null set is a subset of a set

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[InfinityLearn_NEET]

A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.
A set of apples in the basket of grapes is an example of empty set. Because in grapes basket there are no apples present.

[faizulmirza]

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[geliandelacruz]

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[Teteihmar]

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[crazytalks]

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[anurajsingh]

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[binibiningmaica]

That is really Helpful 😌 Can you also have some explanation for truth and falsity using Set?

[rafiqsangat]

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[bishalroy]

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[drmuhammadmemon]

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[raylittlerock]

A better explanation can be found in Enderton, Set theory.

Suppose S is a set ( any arbitrary set).

Now there are 2 and only 2 possibilities: (1) The null set is a subset of S (2) the null set is not a subset of S. RQ : in case (1) is true, (2) is necessarily false, and vice versa.

Saying that a set X is included in a set Y means that all the elements of X are also elements of Y.

With this in mind, let's consider the second alternative, namely (2) above. If (2) were true, it would mean that :

" it is false that all the elements of the null set are also elements of S"

in other words that

" there is at least one object which (i) is an element of the null set and (ii) which is not an element of S"

and, finally, that :

" there is at least one objet which is an element of the null set" .

But of course, this is impossible, since, by definition, the null set is a set which has NO ELEMENT!

The second alternative (2) led us to a contradiction. So logics compells us to admit that (1) must be true: the nul set is a subset ( is included in) our arbitrary set S.

What is true of our arbitrary S set is true for all possible set. ( By the rule of universal generalization)

Conclusion : For all set X, the null set is a subet of X ( in other words, the null set is a subset of every set, included itself).

RQ : another proof can be found using a truth table, and the properties of the " if... then" operator

[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.

[suchithras]

Thank you so much I learn more by watching this video

[tarunkashyap]

Thank you for this video, i was very confused about this topic untile i found this video

[namasteindia]

So can we say that a null set is equal and equivalent to another null set?

[startergod]

Dear mam,
You said ;
A={1, 2, 3}
Where 1, 2, 3 and nothing is present where,
Nothing is a subset of A...
But what about in infinity set ((( like = {infinity..., -1, 0, 1, ...infinity} )) is there "nothing" present ??