Question 467766: Given the following sets, select the statement below that is true.
A= {l,a,t,e,r} B= {l,a,t,e} C= {t,a,l,e} D= {e,a,t}
B is a proper subset of C and C is a subset of A
C is a subset of B and D is aproper subset of B
D is a proper subset of A and A is a proper subset of D
B is a proper subset of A and C is a proper subset of D
D is a subset of A and A is a proper subset of C
Answer by moshiz08(60)   (Show Source):
You can put this solution on YOUR website! 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.
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.
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.
With these definitions in mind, let's go through each answer choice.
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.
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.
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?)
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.
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.
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