Question

For any set A, prove that if A is a subset of the empty set $\left( A\subseteq \varphi \right)$, then prove that A is a subset of that empty set $\left( A=\varphi \right)$.

Answer 1

We start solving the problem by using the fact that an empty set is a subset of every set given in the universe which gives us the relation $\varphi \subseteq A$. We then use the fact that if two sets are subsets to each other i.e., A is subset of B $\left( A\subseteq B \right)$ and B is subset of A $\left( B\subseteq A \right)$, then two sets are equal to each other i.e., $\left( A=B \right)$, which is clearly satisfied by the given two sets to complete the proof.

Complete step-by-step answer:
According to the problem, we are given that A is a subset of the empty set $\left( A\subseteq \varphi \right)$ and we need to prove that A is subset of that empty set $\left( A=\varphi \right)$.
We know that an empty set is a subset of every set given in the universe. Using this fact, we get that the empty set is a subset of the given set A.
So, we get $\varphi \subseteq A$ ---(1).
But according to the problem, we are given that A is a subset of the empty set $A\subseteq \varphi$ ---(2).
We know that if two sets are subsets to each other i.e., A is subset of B $\left( A\subseteq B \right)$ and B is subset of A $\left( B\subseteq A \right)$, then two sets are equal to each other i.e., $\left( A=B \right)$. We can see that this fact is satisfied by the two given sets A and empty set $\left( \varphi \right)$ and we can say that both sets are equal.
So, we get that $A=\varphi$.
So, we have proved that set A is equal to the empty set $\left( \varphi \right)$.
∴ We have proved that if A is a subset of the empty set $\left( A\subseteq \varphi \right)$, then A is a subset of that empty set $\left( A=\varphi \right)$.

Note: We should know that an empty set doesn’t contain any elements in it. So, this tells us that if any set is a subset of an empty set, then it must have zero elements (as containing negative elements in the set is absurd) which is also an empty set. We should not just assume random elements in the empty set as this is wrong and this is the most common mistake done by most of us.