Youtor
The Empty Set
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=EJze-cf1lLs
Check the description below for additional remarks: • 1. I proved ∅ ⊆ S by arguing why ∅ ⊈ S is false. Another way to prove this is to use the definition of a subset directly. ∅ ⊆ S if for all objects x, x ∈ ∅ implies x ∈ S. Since x ∈ ∅ is always false, the implication is vacuously true. Therefore, we can conclude ∅ ⊆ S. • Understanding this proof requires a bit of background in mathematical logic, so I adopted the more intuitive approach in the video, where I show ∅ ⊈ S is false instead. • 2. The empty set is also sometimes called the null set. • 3. The existence of the empty set is sometimes included as an axiom in some formulations of axiomatic set theory. Or its existence can be deduced through something like {x ∈ A | x ≠ x}, provided that the set A exists of course. • 4. In some formulations of mathematics, the number 0 is defined to be the empty set (i.e. 0 := ∅). 1 is then defined to be {0} = {∅}, 2 is {0, 1} = {∅, {∅}}, 3 is {0, 1, 2} = {∅, {∅}, {∅, {∅}}} and so on. • 5. We can prove that the empty set is unique in axiomatic set theory. That is if A is a set with no elements, then it must be the case that A = ∅. • 0:00 - 0:27 Definition • 0:27 - 0:54 Comparing {} and {{}} • 0:54 - 2:17 {} is a subset of any set
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