MHB Sets and basic notation.

[bergausstein]

I have two questions

explain why any subset of a finite set is finite. (prove)

and

why is empty set is considered to be a subset of any set?
I'm confused, because let's say set A is a subset of set B it means that every element of A is an element of B. in the case of empty set being a subset of any set kind of hoodwinked me. please explain.

and also the idea of it being disjoint with any set, even from itself.

 

[Deveno]

Suppose a finite set had an infinite subset. Since EVERY element of the subset is also in the "parent set", this means our original set is also infinite. But that is absurd, we just said it was finite!

Ok, let's see if the empty set is a subset of any other set. All we have to do is check to see if every element of the empty set is in our "given set" (let's call it S).

So, checking now: first element of the empty set is...umm...gee, we don't have any elements! There's nothing to check!

In other words, it is NOT the case that there is some element of the empty set NOT in S, so the "double negatives cancel".

Two sets are said to be disjoint if they have no elements in common. The empty set has no elements, so it cannot possibly have any in common with any other set, even itself!

The way I think of the empty set is just a blank: the contents of an empty container. Surely this "emptiness" is also in every other container (although the containers usually have stuff in them).

Some more confusing stuff:

Every integer in the empty set is even, and also odd! That's because there are exactly NO integers which are both (and a darn good thing, too!). Every element in the empty set is a self-contradiction, so it's really GOOD news that the empty set is empty, or else the universe might just explode.

 

[paulmdrdo1]

for the empty set case consider these:1. The set A is a subset of the set B if and only if every element of A is also an element of B. If A is the empty set then A has no elements and so all of its elements (there are none) belong to B no matter what set B we are dealing with. That is, the empty set is a subset of every set.

2. Another way of understanding it is to look at intersections. The intersection of two sets is a subset of each of the original sets. So if {} is the empty set and A is any set then {} intersect A is {} which means {} is a subset of A and {} is a subset of {}.

3. You can prove it by contradiction. Let's say that you have the empty set {} and a set A. Based on the definition, {} is a subset of A unless there is some element in {} that is not in A. So if {} is not a subset of A then there is an element in {}. But {} has no elements and hence this is a contradiction, so the set {} must be a subset of A.

An example with an empty set and a non-empty set might be this: the (set of all aligators who have walked on the moon) and the (set of all astronauts). Examine the three arguments above with this example in mind
regards!

 

[HallsofIvy]

Just to put in my oar: the definition of "A is a subset of B" is that the statement "if x is in A then x is in B" is true. Further, it is a basis fact of logic that if statement "P" is false, then the statement "if P is true then Q is true" is true whether Q is or not.

So if A is empty, the statement "x is in A" is false for all A. Therefore the statement "if x is in A then x is in B" is true for all x.

If you are not happy with 'if statement "P" is false, then the statement "if P is true then Q is true" is true whether Q is or not' consider the case if, say, A= {1, 2, 3} and B= {1, 2, 3, 4, 5}.

To show that A is a subset of B, we must show that "if x is in A then x is in B" is a true statement, for all x. Certainly, if x is 1, 2, or 3, "x is in A" and "x is in B" are both true so the statement is true. What if x is 4 or 5? Then the statement "x is in A" is false while "x is in B" is still true. Or, what if x= 10? Then "x is in A" is false and "x is in B" is false. Since it is clear that, in this case, A is a subset of B, we need both of those statements to be true. That is why we need ''if statement "P" is false, then the statement "if P is true then Q is true" is true whether Q is or not'.