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#1 2007-05-21 03:43:31

player123
Guest

list all the proper subsets?

hi i have a question about sets. how would you list all the proper subsets of F ?

D = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
E = {2,4,6,8,10,12,14,16,18}
F={2,4}

thanks

#2 2007-05-21 04:39:43

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: list all the proper subsets?

They are just three of them: Ø, {2}, {4}.

A proper subset of a set X is any subset of X that is not X itself. This includes the empty set if X is not the empty set.

Last edited by JaneFairfax (2007-05-21 07:33:04)

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#3 2007-05-21 04:50:11

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: list all the proper subsets?

Are you sure? I was taught that proper subsets didn't include the empty set. I suppose it doesn't really matter as long as you don't switch definitions during a proof or something.

With sets as small as that, it's easy to see what all the subsets are, but if you have a larger one, then it can help to write out binary numbers and assign each digit to an element of your set.

For example, for the set {1,2,3}, you would write out the first 8 binary numbers and then map a subset to each one.

000 --> :phi
001 --> {3}
010 --> {2}
011 --> {2,3}
100 --> {1}
101 --> {1,3}
110 --> {1,2}
111 --> {1,2,3}

In general, for a set of size n, you'd need to write the first 2^n binary numbers.


Why did the vector cross the road?
It wanted to be normal.

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#4 2007-05-21 07:31:52

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: list all the proper subsets?

Typically, "subset" is equated with "less than or equal to".

This can be illustrated by the following property:

Similarly with numbers, if a ≤ b and b ≤ a, then a = b.

In a similar manner, proper subset is equated with "less than", where A is a subset of B but A does not equal B.  Following this line of thinking, the null set is a proper subset of any non-empty set.

The other line of thinking is that "proper" means "non-trivial".  Since every set S is a subset of S, S is considered trivial.  Since the null set is a subset of S for every S, the null set is considered trivial.

It really doesn't matter which you choose, just make sure whoever you're talking to or doing assignments for is using the same definition as you are.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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