Question

Help me to prove this .. The empty set is a subset of every set.
Thanks

Answer 1

Well if you make E (everything possible) value or setwise then the null set is included in everything:

∅ ⊆ E

Then for any set or subset A in E you would have either

∅ ⊂ A or ∅ ⊂ A' since it has to be in one of these since the complement of A, which is A' is everything else in E not in A. This is obvious.

Let's assume the second always, that the null set is a subset of the complement of any given set and not a part of set we look at. Then for A the nullset is located in A'

But then let's look at a set B which is A', and if we follow our rule we assume then the nullset is located in B' which is A. Then from this method we can see that the nullset is a subset of both A and B and A' and B'. It is obvious from this point that any given set has ∅ as a subset.

Q.E.D.

Answer 2

To prove that a null set is a subset of all sets.

Proof :

We know that for any set A , A&B belongs to A.

Now Let B be the null set. Then, A & null set belongs A.

But A & null set = null set.

Therefore ,the null set belongs to A.

A null set has the property: (i) A Union Null set = A (ii) A intersection Null set = Null set.

Null set is similar to zero in property : For any number n and zero n*0 = 0 and n+0 = n


A null set has no element. So under the definition of subset , every element of the null set belongs to set A is equivalent to no element of the null set belongs to A.

Answer 3

thank you very much :))))))

Answer 4

u r most welcome