# How do you prove that the null set is a subset of all sets?No.

hustoncmk | High School Teacher | (Level 1) Assistant Educator

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The null set is an empty set, and includes nothing within the brackets.  Every set that has at least one element, has a subset that is lacking any elements, also known as the null set.

Let's look at the set of numbers that includes 1, 2, & 3.

It would be written {1,2,3}

subsets would be {1,2}, {1,3}, {2,3}, {1}, {2}, {3}, and {}.

Let's look at this from a completely different perspective.  You have a fresh baked apple pie.  The pie (set) is divided up (subsets) and eaten.  You are left with an empty pie plate.  The pie plate represent the brackets and the emptiness is your nul set.

I hope this helps explain.

neela | High School Teacher | (Level 3) Valedictorian

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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.