Question 467766
Recall that two sets are equal if they have exactly the same elements. In your example, B and C are equal because the elements are the same letters a,e,l,t. 
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We say that X is a subset of Y if everything that is in X is also in Y. We say that X is a proper subset of Y if X is a subset of Y which is not equal to Y. 
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Note that if X equals Y, then X is a subset of Y because everything in X must be in Y. Similarly Y is a subset of X because everything in Y is also in X. In other words, X = Y if and only if X is a subset of Y and Y is a subset of X. 
<br>With these definitions in mind, let's go through each answer choice. 
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 B is a proper subset of C and C is a subset of A
-FALSE. It is true that C is a subset of A. But B is equal to C, so it is not a PROPER subset. 

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 C is a subset of B and D is aproper subset of B
-TRUE. C is a subset of B because C=B, so everything in C is also in B. D is a proper subset of B because everything in D is also in B, but they are not equal. 

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 D is a proper subset of A and A is a proper subset of D 
-FALSE. D is a proper subset of A, but A is NOT a proper subset of D. (In fact, A is not even a subset of D because it has elements l and r which are not in D. In general if X is a proper subset of Y, then Y cannot be a proper subset of X. Can you see why?) 

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 B is a proper subset of A and C is a proper subset of D
-FALSE. B is a proper subset of A, but C is NOT a proper subset of D because there is an element l in C which is not in D. 

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 D is a subset of A and A is a proper subset of C
-FALSE. D is a subset of A, but A is NOT a proper subset of C. As in the third choice, A is not even a subset of C because there is an element r in A which is not in C.