If set A is a proper subset of setB and set B belongs to setC then will set a belong to set C?

it is given in a book that a will not belong to c .but how?


alsi for a set to belong to some other set is it essential that all elements of one set should be present in the other.

If set A is a proper subset of setB and set B belongs to setC then will set a belong to set C?
Not necessarily. A note to the above answerer: "belongs to" is not ambiguous in any way. If x belongs to X, then x is an element of X. It's fairly standard terminology.





So here's the problem:





C is a set of sets. The set B is in C.





The set A might be in C. Is it?





A=B is not an option, however. Because A is a proper subset of B. There is no rule that says if C contains B, then C must contain all of its proper subsets.





Thus the answer is no (not necessarily).





Example: C is the set of {my cat, my marbles, my TV}





B = my marbles





A = my green marbles





C does not contain "my green marbles" - even though it contains "my marbles" because we're not talking about which marbles it contains, we're talking about two distinct sets of marbles.
Reply:I think what the book is trying to point out is the loose terminology in the question.


If the question said:


" If A is a proper subset of B and set B IS A PROPER SUBSET OF C then will A BE A PROPER SUBSET of C?"


This is a true statement.


Being a "subset of" another set is different from "belonging to" a set.


Consider the following example:


A = {1, 2, 3}


B = {1, 2, 3, 4, 5}


and C = { {1, 2, 3, 4, 5}, {1, 2} }


Certainly A is a proper subset of B, each containing numbers.


C, on the other hand, is a set of sets. It contains B, but the set A is not contained in set C.


Does this help?