2. Empty set / Paradoxes of set theory // Ivan Yashchenko ∀ ∀ x, y, z

1. The similar

What does that mean is a subset of the set ? This means that all elements of the set belong and set . If you imagine sets in the form of boxes, then - this is a big box, and many - a smaller box, in which some of the elements lying in the box lie . Designation: .

For example, the set of all even numbers is a subset of the set of all integers, and the set - a subset of the set .

Consider two sets:

all flying crocodiles and all participants of the Olympiad .

Is one of them a subset of the other?

How to prove that ? You can check that any item sets lies in . And you can apply the method by contradiction * 2 : if a not a subset then there is an element such that and if so no then .

But is it possible to find a flying crocodile, not participating in the Olympiad? But where can you even find a flying crocodile ... Therefore

all flying crocodiles all participants of the Olympiad * 3 .

The set of flying crocodiles is an empty set: there are no elements in it. This set is so important that for it even came up with a special symbol: *four . There is only one symbol for the empty set, because the empty set is unique. Indeed, suppose that there are two different empty sets. But what means that the sets are different? This means that in one of them there is an element that does not belong to the other. But in empty sets there are no elements at all!

So, we have proved that the empty set is unique and is a subset of any other set.

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