What is the difference between subset and proper subset?

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In summary, my book defines a proper subset: "a Set A is a proper subset of a set B if A \subseteq B but A \neq B." If A is a proper subset of B we write A\subsetB.
  • #1
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My book defines a proper subset: "a Set A is a proper subset of a set B if A [tex]\subseteq[/tex] B but A [tex]\neq[/tex] B. If A is a proper subset of B we write A[tex]\subset[/tex]B."

For example, S={4,5,7} and T={3,4,5,6,7}, then S [tex]\subset[/tex] T.


So, from my understanding, every element in S is contained in T however there is at least one other element in T not contained in S.

So what would an example of A[tex]\subseteq[/tex]B be?

My text says the [tex]N[/tex][tex]\subseteq[/tex][tex]Z[/tex] (Natural numbers and integers, respectively).

But every element of [tex]N[/tex] is contained in [tex]Z[/tex] and they are not equal, so wouldn't we write [tex]N[/tex][tex]\subset[/tex][tex]Z[/tex] ?


What would be an example of three sets A,B,C such that A[tex]\subseteq[/tex] B and B [tex]\subset[/tex] C ? (the notation is coming out funny looking for some reason... "A is a subset of B and B is a proper subset of C", is what I'm trying to say.

Would this be correct...

A={1,2,3}, B={1,2,3}, C={1,2,3,4} ?


Or would this be correct...

A={1,2,3}, B={{1,2,3}}, C={{1,2,3},4}
 
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  • #2
Almost every book I encountered always uses the "[tex]\subseteq[/tex]" notation, except in situations where it's important to emphasize that some set is a proper subset of another. You can look at the "[tex]\subseteq[/tex]"-notation as a more general one - in most situations, the set of interest may or may not be a proper subset of another set, both situations are possible and don't affect the result.
 
  • #3
radou said:
Almost every book I encountered always uses the "[tex]\subseteq[/tex]" notation, except in situations where it's important to emphasize that some set is a proper subset of another. You can look at the "[tex]\subseteq[/tex]"-notation as a more general one - in most situations, the set of interest may or may not be a proper subset of another set, both situations are possible and don't affect the result.

Did you see the last part of my post? I'm trying to find an example (specific) of sets A, B, and C as described above.
 
  • #4
Your example A={1,2,3}, B={1,2,3}, C={1,2,3,4} is correct for the relation you're trying to give an example of. Or, you can take A = N, B = Z, C = R. Or, A = Z, B = Z, C = R.
 
  • #5
radou said:
Your example A={1,2,3}, B={1,2,3}, C={1,2,3,4} is correct for the relation you're trying to give an example of. Or, you can take A = N, B = Z, C = R. Or, A = Z, B = Z, C = R.

Why would N[tex]\subseteq[/tex]Z instead of N being a proper subset of Z?

Clearly they are not equal, meaning that Z has more elements.

(for some reason the symbol for proper subset is just not working!... the elongated "C" without the slash on the bottom). maybe it's my browser?
 
  • #6
The point is, whatever relation you use on N and Z respectively, the result is always true. It is true that N is a proper subset of Z. And it is true that N[tex]\subseteq[/tex]Z, too.
 
  • #7
radou said:
The point is, whatever relation you use on N and Z respectively, the result is always true. It is true that N is a proper subset of Z. And it is true that N[tex]\subseteq[/tex]Z, too.

But N is a proper subset of Z since Z contains, among other extra elements 0.

So the symbol "C" with a dash below should not work to relate them, according to my book.


(jesus, this thing isn't letting me insert any characters anymore, not even if I copy and paste from yours!)
 
  • #8
How does your book define [tex]\subseteq[/tex] then??

Both [tex]N\subset Z[/tex] and [tex]N\subseteq Z[/tex] are correct. It's the same as the difference between 3<5 and [tex]3\leq 5[/tex]. They are both correct. However, the latter is more general: you can write [tex]5\leq 5[/tex] but not 5<5.
 
  • #9
micromass said:
How does your book define [tex]\subseteq[/tex] then??

Both [tex]N\subset Z[/tex] and [tex]N\subseteq Z[/tex] are correct. It's the same as the difference between 3<5 and [tex]3\leq 5[/tex]. They are both correct. However, the latter is more general: you can write [tex]5\leq 5[/tex] but not 5<5.

I see... it just "clicked"... the dash on the bottom doesn't imply equality, it only implies the possibility that the two sets are the same... does that make sense?

Also, what would be an example then of my A,B,and C sets listed in my first post?
 
  • #10
Yes, I believe you've got it now. Your example A={1,2,3}, B={1,2,3}, C={1,2,3,4} is correct. But also A={1,2}, B={1,2,3}, C={1,2,3,4} would be correct...
 
  • #11
micromass said:
Yes, I believe you've got it now. Your example A={1,2,3}, B={1,2,3}, C={1,2,3,4} is correct. But also A={1,2}, B={1,2,3}, C={1,2,3,4} would be correct...

ok thanks, and thank you everyone who replied, much appreciated.
 

1. What is the difference between a subset and a proper subset?

A subset is a set that contains all the elements of another set, and may also contain additional elements. A proper subset, on the other hand, is a subset that contains all the elements of another set, but does not contain any additional elements.

2. Can a set be both a subset and a proper subset of another set?

Yes, a set can be both a subset and a proper subset of another set. This occurs when the set in question is equal to the other set, making it both a subset and a proper subset.

3. How can you tell if a set is a proper subset?

A set is a proper subset if it is a subset but not equal to the other set. This means that there must be at least one element in the other set that is not in the subset.

4. What is the significance of understanding the difference between a subset and a proper subset?

Understanding the difference between a subset and a proper subset is important in mathematical reasoning and proof. It allows us to accurately define and compare sets, and determine relationships between them.

5. Are all proper subsets also subsets?

Yes, all proper subsets are subsets, but not all subsets are proper subsets. A proper subset is a specific type of subset that does not contain all the elements of the other set.

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