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

[barneygumble742]

hi,

if A = {1, 2, 4, 6} and B = {0, 2, 3, 4, 5}

my understanding is that 1 is not a subset of A and 0 is not a subset of B. however 1 is an element of A and 0 is an element of B.
---

also if A = { {1}, 2, {4}, 6} and B = { {0}, 2, {3}, 4, 5}

then i can say that 1 is a subset of A and 0 is a subset of B?
---

if A = { {1}, 2, {4}, 6} and B = { {0}, 2, {3}, 4, 5}
i can say that 4 is a proper subset of A and that 3 is a proper subset of B?
---

i thought i knew this stuff until i got my first quiz back...not good.

thanks.

 

[Muzza]

if A = {1, 2, 4, 6} and B = {0, 2, 3, 4, 5}

my understanding is that 1 is not a subset of A and 0 is not a subset of B. however 1 is an element of A and 0 is an element of B.

Correct.

also if A = { {1}, 2, {4}, 6} and B = { {0}, 2, {3}, 4, 5}

then i can say that 1 is a subset of A and 0 is a subset ofB?

No. However, you may say that { {1} } is a subset of A and { {0} } is a subset of B. You could also say that {1} is an element of A, etc.

"Being a subset of" is a relation defined on /sets/, i.e., we can only make sense of the phrase "X is a subset of Y" if X and Y are both sets.

Are 1 and 0 sets?

 

[barneygumble742]

if A = { {1}, 2, {4}, 6} and B = { {0}, 2, {3}, 4, 5}

in that statement, aren't {1} and {0} sets?

---

also more importantly, this is what i have:
A = {1, 2, 3}
B = {0, 2, 4}
C = {1, 2, 3, 4, 5, 6}
D = { {0, 2, 4} }

1. 1 is NOT a proper subset of A
2. 1 is NOT a subset of A
3. A is NOT an element of A
4. A is NOT a proper subset of A
5. 0 is NOT a proper subset of B
6. 0 is NOT a subset of B
7. A is NOT a subset of B
8. B is NOT an element of D

i'm sorry if it's long but i just can't picture it in my head. I've gone over the definitions many times but i just can't get the idea of what's a subset and what's a proper subset.

thanks.

 

[Muzza]

if A = { {1}, 2, {4}, 6} and B = { {0}, 2, {3}, 4, 5}

in that statement, aren't {1} and {0} sets?

Yes, but all that means is that it makes sense to actually ask the question "is {1} a subset of A". It still doesn't make {1} a subset of A, since that would require that 1 be an element of A...

8. B is NOT an element of D

That's the only one that I can see is wrong. B is an element of D.

 

[barneygumble742]

could you please show me an example of a subset and a proper subset?

 

[Muzza]

{1, 2, 3} is a subset of {1, 2, 3} (but it's not proper). {1} is a proper subset of {1, 2, 3}.

 

[barneygumble742]

that sounds too simple.

A = {1, 2, 3}
B = {0, 2, 4}
C = {1, 2, 3, 4, 5, 6}
D = { {0, 2, 4} }

from the above sets,
1 is subset of A and 1 is also a proper subset of A.
A and B are subsets of C but only A is a proper subset of C.
B is a subset of D. B is also a proper subset of D. B is an element of D.

thanks

 

[Muzza]

1 is subset of A and 1 is also a proper subset of A.

No. Do you not know the difference between 1 and {1}?

A and B are subsets of C but only A is a proper subset of C.

B is not a subset of C. Do you know the definition of "subset"?

B is a subset of D. B is also a proper subset of D.

No. If B were a subset of D, then the elements of B would be in D, so for example, 0 would be in D (but it isn't - 0 is contained in a set which is an element of D, but that's different).

B is an element of D.

Yes.

 

[matt grime]

Again you're confusing subset and element.

A is a set. it's elements are 1,2,3.

The subsets of A are sets whose elements are also elements of A, right?

so {1,2} is a subset of A.

B is not a subset of C since 0 is an element of B but not an element of.

B is not a subset of D. B is an element of D. D is a set that contains exactly one element. That element is the set {0,2,4}

 

[barneygumble742]

not to sound ignorant because my professor is ineffective.

is there a difference between 1 and {1}?

A = {1, 2, 3}
B = {1, 2}
C = { {1, 2, 3} , 4, 5, 6}

in the above sets, is A a subset of C and B a proper subset of C?

i have a horrible teacher and an awful book. i think i can better understand this better if there was a book that visually explains it. is there a book you can recommend? i think i might have to teach myself this material.

 

[matt grime]

Do not blame the teacher: you'll get no sympathy from the many of us that are teachers and all of whom have students who moan that you're ineffective. I assure you we have equally strong opinions about you too. If you learn the definition of set, element and subset, then all of these questions are self explanatory.


Curly braces denote a set, what inside the curly braces are elements of that set. Thus {1} is a set that contains one element, that element is 1, whatever that may be.

Neither A nor B are subsets of C. A is an element of C. B is a (proper) subset of A

 

[barneygumble742]

i've never blamed a teacher/book ever before. but moreso, this person is truly condescending of students who approach him during his office hours. so its the person, not the profession.

the last statement you said pretty much makes it all clear and i understand that. I'm going to print this page. can you as a teacher, recommend a book with plenty of examples?

 

[matt grime]

No I cannot recommend a textbook for this. I, as a teacher, consider this to be elementary and not something that a book is required for since it should have been explained in High School and I work at a university. This is not a dismissal of your problem but a reflection of my ignorance.

Note it is possible for a set to contain another set as an element. This happened in C in the last question. That other set is merely an element it is not a subset. A set containing this other set is a subset.

 

[HallsofIvy]

Warning: Some texts use the word "proper" subset of A for any subset of A other than A itself. Other texts also deny the "proper" adjective to the empty set. Be sure you know which your textbook or class is using!

 

[honestrosewater]

I think these examples should straighten things out. Have your definitions handy:
A set S is a subset of set T iff every member of S is also a member of T.
A set S is a proper subset of T iff S is a subset of T and there is some (at least one) member of T that isn't a member of S.

T = {{{}}}. Which of the following are subsets of T? Proper subsets?
A. {{{}}}
B. {{}}
C. {}

T = {{}, {{{}}}}. Which of the following are subsets of T? Proper subsets?
A. {{}, {{{}}}}
B. {{{{}}}}
C. {{{}}}
D. {{}}
E. {}

 

[barneygumble742]

C and B are subsets and C is a proper subset?

D and E are subsets and D and E are proper subsets?

A set S is a subset of set T iff every member of S is also a member of T.
A set S is a proper subset of T iff S is a subset of T and there is some (at least one) member of T that isn't a member of S.

---

my understanding is that a set S is a proper subset of T iff every element of S is also a member of T. And a set S is a subset of T if one or more (but not all) elements of S are in T.

thanks

 

[honestrosewater]

my understanding is that a set S is a proper subset of T iff every element of S is also a member of T. And a set S is a subset of T if one or more (but not all) elements of S are in T.

Do you have the definitions you were given? These definitions are standard; They should be basically the same as the ones I gave. Can you try the examples again with the definitions I gave?

 

[barneygumble742]

the definitions I'm given are:
Let A and B be sets.
A is a subset of B if every element of A is also an element of B.
A is a Proper Subset of B if A is a subset of B but A is not equal to B.
---
from the defintion of subset, i take it that A has to equal B for B to be a subset of A.
and for a proper subset, B has to have the same elements as A but B can have more elements as well.

 

[honestrosewater]

the definitions I'm given are:
Let A and B be sets.
A is a subset of B if every element of A is also an element of B.
A is a Proper Subset of B if A is a subset of B but A is not equal to B.
---
from the defintion of subset, i take it that A has to equal B for B to be a subset of A.
and for a proper subset, B has to have the same elements as A but B can have more elements as well.

Right. If A is a subset of B and B is a subset of A, then A = B. If A is a subset of B, but B is not a subset of A, then B must have an element that isn't in A. Do you understand it all now?

 

[barneygumble742]

yes. thank you very much.

i just got out of class and after class i spoke to my professor who made it clear.

...just takes me more time than for others.

 

[Avaloyuru]

I have been following this particular post and need a bit more clarification. I am in a college level math class for the first time after being out of school for over 35 years. I believe that I am beginning to understand however I just want to make sure of a few things.

If you have a set as in A = {1, 2, 3, 4, 5, 6} and then you have other sets such as B = {2, 4, 6} and C = {1, 3, 5} then both B and C are considered Subsets of A because all of the elements of both B and C can be found in A.

However, if you have another set say D = {1, 3, 5, 7} then D is considered a Proper Subset because even though it contains the additional element of {7}, all of the other elements {1}, {3} and {5} are also contained in Set A.

 

[gb7nash]

If you have a set as in A = {1, 2, 3, 4, 5, 6} and then you have other sets such as B = {2, 4, 6} and C = {1, 3, 5} then both B and C are considered Subsets of A because all of the elements of both B and C can be found in A.

Correct

However, if you have another set say D = {1, 3, 5, 7} then D is considered a Proper Subset because even though it contains the additional element of {7}, all of the other elements {1}, {3} and {5} are also contained in Set A.

No. D is not a proper subset of A since D contains an element that A does not contain (namely 7). In order for D to be a proper subset of A, every element of D must be in A, and D must not be equal to A.

 

[nickbyfleet]

i've never blamed a teacher/book ever before. but moreso, this person is truly condescending of students who approach him during his office hours. so its the person, not the profession.

the last statement you said pretty much makes it all clear and i understand that. I'm going to print this page. can you as a teacher, recommend a book with plenty of examples?

I am not a teacher, rather a Math major at a pretty decent uni. This material is covered very well in Introductory "Mathematics: Algebra and Analysis by Geoff Smith (Springer)" and is a great introduction to the field of pure mathematics. Hope that helps. Cheers ~Nick