Question 481352
&#8838 means improper subset
&#8834 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 &#8838 B and B &#8838 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 &#8834 B is valid.
B &#8834 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.
<a href = "http://answers.yahoo.com/question/index?qid=20080125081016AA94Fwe" target = "_blank">http://answers.yahoo.com/question/index?qid=20080125081016AA94Fwe</a>
i analyzed each of your statements in turn and found the following:
<pre>
1. E &#8838; A and B &#8834; 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 &#8834; D and E &#8834; 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 &#8834 C would be more appropriate.
3. D &#8838; C and D &#8838; 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 &#8834; E and B &#8838; A
     C is not a subset of E because E contains fewer elements than C.
5. D &#8834; C and B &#8838; 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 !!!!!
</pre>
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 &#8838 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 &#8834 C is correct.
I'd go with selection 5.