Question

what is the empty set? Show that the empty set is a subset of every set.

Short answer

The empty set is the set that does not contain any elements$\cancel{O}$ is a subset of every set.

Step-by-step solution

Definition of set 

The empty set is the set that does not contain any elements

Notation:$\varnothing$

Empty set is the subset of every set.

To proof: $\varnothing \subseteq A$, for every set A.

Proof:

Let A be a set.$\varnothing$ is a subset of A if all elements of $\varnothing$are also elements of A:

$\forall x(x\in \varnothing \to x\in A)$

Since, $\varnothing$does not contain any elements, the statement $x\in \varnothing$is false.

$\equiv \forall x(F\to x\in A)$

Use logical equivalence (1):

$\equiv \forall x(\neg F\vee x\in A)$

The negation of false is true

$\equiv \forall x(T\vee (x\in A\left)\right)$

Use the domination law

$\equiv \forall x\left(T\right)$

Since T does not dependent on

$\equiv T$

Thus, we have shown that the statementis$\forall x(x\in \varnothing \to x\in A)$ always true for every set A and thus $\varnothing$is a subset of every set.