What Is the Difference Between Proper and Improper Subsets? Examples Included!

[KIX369]

I need some help here ...

I won't to know what the difference between the proper and improper subset ... with examples ...

 

[CompuChip]

If you check the definition of a subset of some set X, you will find that also the empty set {} and X itself satisfy it.

However, our intuition often says that a subset should really be "smaller" than X, for example, {1, 2, 3} should not be a subset of {1, 2, 3}, but {1, 2} should.

This is what we try to capture in "proper subset". Mathematically, it is a non-empty subset not equal to the whole set.

So {1, 2, 3} and {1, 2} are both subsets of {1, 2, 3}, but only {1, 2} is a proper subset

 

[KIX369]

thank you Mr. Compuchip ...


but what about improper subset ...

 

[CompuChip]

I never used that word.
But apparently S is an improper subset of X if and only if S = X.
(Personally, I would then also include the case of S being empty here).

 

[KIX369]

thank you again Mr . Compuchip ... thank you so much ...

 

[HallsofIvy]

Some texts (I think the majority) use the phrase "proper subset of A" to mean any subset of A other than A itself. Other texts use the phrase to mean any subset of A other than A itself or the empty set.

 

[Mensanator]

In another context, the set of divisors of an integer are called the aliquot parts, where the divisors are said to be "proper", i.e., the set does not include the number itself.

Thus, the aliquot parts of 12 are {1,2,3,4,6}. 12 is not included as it is not a "proper" divisor.

 

[KIX369]

thank you again so much