Question

If A = \left\\{ {0,2,4} \right\\} find $A \cap \phi$ and $A \cap A$

Answer 1

First, we understand the definition of the intersection of two sets. Now use the properties of intersection. Then give set $A$ to the intersection with an empty set. Then set $A$ to the intersection with itself. Finally, we get an answer.

Complete step-by-step answer:
The given set $A$ is nonempty. Set A = \left\\{ {0,2,4} \right\\}
First, we understand the intersection meaning $A$ and $B$ are sets. The common elements or objects of the two sets form the set. That is called $A$ intersection $B$
A \cap B = \left\\{ {x:x \in A\,and\,x \in B} \right\\}
This is the definition of the intersection of two sets.
Now first we find $A \cap \phi$
Set $A$ is \left\\{ {0,2,4} \right\\}. $\phi$ means an empty set or null set. The set has no elements.
So that \phi = \left\\{ {} \right\\}
Now we find the intersection of $A$ an empty set.
A \cap \phi = \left\\{ {} \right\\}
Because the common element of both sets is none. So that answer of $A$ intersection empty set is a null set.
Next, we find $A \cap A$ value,
We know that $A$ value is \left\\{ {0,2,4} \right\\}
Set $A$ is the intersection to itself. That means A \cap A = \left\\{ {0,2,4} \right\\}
Because the common element of both sets is whole elements. It means the intersection value $A$ and $A$ is $A$.
So that,

A \cap A = \left\\{ {0,2,4} \right\\} \\\ = A \\\

The intersection of the empty set to non-empty is always an empty set. And the intersection of a set with itself is always getting the same set. These are important properties of the intersection of two sets.
Finally, we get the answers
A \cap \phi = \left\\{ {} \right\\},\,A \cap A = A

Note: First given a question to understand the definition of intersection or union of two sets. The intersection of the empty set to non-empty is always an empty set. And the intersection of a set with itself is always getting the same set. These properties are used carefully. Don't confuse intersection and union properties. Because union property is changeable.