Why is A intersection A' null?

Question

Intersection between 2 sets means the elements that are common to both those sets. I read that at Wikipedia.

$\varnothing$ is the subset of every set but not an element of all the sets.

If $\varnothing$ is not an element of every set, then why is A intersection A' = null?

Edit: I just started reading sets. I had this doubt and I asked it. Sorry I might have done something wrong but please don't be angry at me.

Suppose A = {1,3,5} and U = {1,2,3,4,5} So A' = {2,4}. So why will A intersection A' = null, when null is not an element of any of the two sets (A and A')?

Answer 1

See the definition of Complement (of a set), denoted : $A^c,\overline A,A′$.

If $A′$ is the complement of the set $A$, obviously $A \cap A′= \emptyset$, because from $x ∈ A \cap A′$ we have that $x∈A$ and $x∈A′$.

The last one is equivalent to $x∉A$, and thus we have bot $x∈A$ and $x∉A$ : contradicition !

Thus there are no common elements to $A$ and $A'$, i.e. their intersection $A \cap A'$ is empty.

Two sets are equal exactly when they have the same elements.

$\emptyset$ is empty: i.e. it has no elements.

We have shown that $A \cap A′$ has no elements (because there are no common elements to a set and its complement).

Thus, the two sets: $A \cap A′$ and $\emptyset$ are equal, because both have no elements.

Answer 2

I don't have enough reputation yet, otherwise I'd add this as a comment.

Subsets are not the same things as elements, so although $\emptyset$ is a subset of both $A$ and its complement, whether or not it is an element depends on what exactly is contained in $A$. If it is an element of $A$ it will not be an element of $A$'s complement.

Related: Is empty set element of every set if it is subset of every set? and Is the null set a subset of every set?