Question

how to prove null set is subset of every set

Answer 1

When is A a subset of B?

Answer 2

A is subset of B if every element of A is also the element of B.

Answer 3

Right, so is that true for the null set and some set A?

Answer 4

i am unable to say i am confused , it is to be proved theoretically.

Answer 5

Just to clarify, by the null set you mean the empty set, right?

Answer 6

yes

Answer 7

Well every element in that set is in A. There are no elements in that set, so there's nothing to prove.

Answer 8

actually it is asked in the exam we have to prove it by set builder notations

Answer 9

Ok, so A is a subset of B if and only if \[A \cap B=A\]

Answer 10

Is that true for the null set and some arbitrary set?

Answer 11

i don't think it is true for null set as it has no elements , above relation is true with non-empty set.

Answer 12

\(A\cap B\) is the set of all elements in both A and B. So what do we have for \(\emptyset \cap A\)?

Answer 13

that will be empty set but can we say from that ,null set is subset of every set i am not sure.

Answer 14

That's what I said earlier: A is a subset of B if and only if \(A \cap B =A\). But let's prove it if you don't believe it.

Answer 15

Let's prove only from 'from left to right', that's the part we need. So let's assume \(a \cap B =A\) then we need to show that A is a subset of B.

from \(A \cap B = A\) we know that A is the set of all elements in both A and B. So every element in A is in B. So A is a subset of B.

Answer 16

Now, A can be anything so also the nullset. And it follows that the null set is a subset of B. any B.