Subset vs Proper Subset

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  • #1
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Main Question or Discussion Point

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}
 

Answers and Replies

  • #2
radou
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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
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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
radou
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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
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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
radou
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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
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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
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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
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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
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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
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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.
 

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