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Sets

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> If every element of A is present in Set B, then A is a subset of B

Yes, this is exactly the definition of a subset.

> Does this hold true only when the cardinalities of A and B are the same?

No.

> If A = {1,2,3} and B = {2,1,3}, can I say that A and B are equal sets?

Yes.

> If so, does it mean that the order of the element in the set does not matter and only the existence of that particular element does?

I would phrase this even more strongly: there is no ordering of the elements in a set at all. Of course when we write the set down we have to choose some ordering, but that ordering isn't inherent to the actual set in any way. Really {1,2,3} and {2,1,3} are just different names for the same mathematical object, just as 1/2 and 2/4 are different names for the same object.

> if A is a proper subset of B, can we say (without looking at the elements of the sets themselves) that n(A) < n(B)?

As long as A and B are finite, yes.

> "Let A be a set then the null set is a subset of A." I can't seem to wrap my head around this; Google says it's an axiom and axioms can't be proven.

It is not an axiom, but rather follows from the definition of a subset. The empty set has no elements, and so trivially all of those elements are also elements of any other set A that we are comparing it to. For intuition, it may help to think of subsets in the alternate but logically equivalent sense: a set A is *not* a subset of the set B if there is an element of A which is not in B.
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1. Yes if every element of A is also in B, then A is a subset of B. This doesn't depend on the cardinality of A and B. For example {1,10} is a subset of the natural numbers {1,2,3...}, because 1 and 10 are both natural numbers

2. Yes

3. Not sure what you mean with n(A)? If you are thinking of cardinality, then the answer is no. For example the even natural numbers {2,4,6...} are a proper subset of the natural numbers, but they have the same cardinality (because we have a bijection from the naturals to the even by f(n)=2n)

4. Yes the empty set is subset of every set. This is something we call vacuous true: if the empty set was not a subset of A, then there would exists an element in the empty set which is not in A. However in the empty set there doesn't exists any elements. Therefore the empty set must be a subset of A
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1. Yes. This is the definition of a subset. Immediate corollaries are that A is a subset of itself (we sometimes make the distinction between Subset and Proper Subset) and that you can have infinite sets with the same cardinality without being the same set. For example, **N** and **Z** have the same cardinality, **N** is clearly a subset of **Z**, but they are not the same sets.
2. Yes. Order of elements does not matter in defining a set. You're bang on the money with this one.
3. No. As mentioned in example 1, the set of natural numbers is a proper subset of the set of integers. All elements of **N** are in **Z** but there are elements of **Z** not in **N**. However they still have the same cardinality. If both sets are finite, however, then it *would be true* to say one has a strictly smaller cardinality than the other.
4. The null set has no elements. Therefore, for any given set S, there are no elements of the null set that aren't in S. Trivially, this means the null set is a subset of S. (This is very similar to something being vacuously true, if you're familiar with that term).

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