⊆ means improper subset
⊂ means proper subset
an improper subset means that A is a subset of B, but B doesn't contain any elements in it other than the elements that are also in A.
an example would be:
A = {a,b,c,d}
B = {a,b,c,d}
these sets are identical so:
A ⊆ B and B ⊆ A are valid.
a proper subset means that A is a subset of B, but B contains additional elements that are not in A.
an example would be:
A = {a,b,c}
B = {a,b,c,d}
A ⊂ B is valid.
B ⊂ A is not valid.
All elememnts in A are in B, but B contains additional elements not in A, namely d.
here's a reference from the web that explains it as well.
http://answers.yahoo.com/question/index?qid=20080125081016AA94Fwe
i analyzed each of your statements in turn and found the following:
1. E ⊆ A and B ⊂ C
E is a proper subset of A because it has less elements than A.
B is an improper subset of C because they both contain the same elements.
This answer is not correct.
2. C ⊂ D and E ⊂ C
C cannot be a subset of D because D contains fewer elements than C.
if anything, it would be the other way around.
D ⊂ C would be more appropriate.
3. D ⊆ C and D ⊆ E
D is a proper subset of C, not an improper subset.
Also D is not a subset of E because E contains fewer elements than D.
4. C ⊂ E and B ⊆ A
C is not a subset of E because E contains fewer elements than C.
5. D ⊂ C and B ⊆ C
D is a proper subset of C because D contains fewer elements than C.
B is an improper subset of C because they both contain the same elements.
THIS ONE LOOKS CORRECT !!!!!
I believe your answer is selection 5.
D contains {b,l,a}
C contains {b,l,a,e}
B contains {b,l,a,e}
it does help to reorder the terms so you can see the relationships easier.
you can see that B and C are identical sets, so B ⊆ C is correct.
you can see that all elements in D are also in C and that D contains fewer elements than C, so D ⊂ C is correct.
I'd go with selection 5.