empty-set

The empty set

https://en.wikipedia.org/wiki/Empty_set

• The empty set is a set without elements, its cardinality is 0

• it is denoted by {} also ∅ (latex: \varnothing $\varnothing$)

• it is a unique set (hence the empty set)

• many possible properties of sets are vacuously true for the empty set

Set is primitive mathematical notion - it is not defined in terms of any previously defined concepts. It escapes any attempt to formally define it, so it must be taken for granted, usually by appealing to one's intuition.

The empty set is also a primitive notion. To state that it exists would require an implicit axiom.

The empty set contains no elements, nevertheless it is regarded as a mathematical object in its own right, once defined within a theory. It may be introduced as a mathematical primitive or its existence may be assured by an axiom, e.g. the axiom of the empty set. In other theories, the existence of the empty set may be derived from the axioms (since it's handy it's useful to derive it early and use it as a lemma thereafter).

The empty set

• There is only one empty set (hence the empty set).

• The empty set is subset of every set

• The empty set is a member of a set's powerset.

• The empty set is not a member of a set's partition.

The empty set is a subset of every set

$$\varnothing \subseteq S \\ \varnothing \subseteq \varnothing \\ \varnothing \subseteq \mathcal{U} \\$$

existing object, usually denoted by $\varnothing$, so $\{\}=\varnothing = \emptyset$

The empty set is the only set whose cardinality is zero, $|\varnothing|=0$.

$$\overline{A\cup B} \\ \varnothing = \{x:x\not\in X\}$$

The empty set is a subset of any set.

$$\varnothing \subseteq X , \ \ \text{where X is any set}$$

The empty set is a proper subset of any non-empty set.

$$\varnothing \subset X , \ \ \text{where X is a non-empty set}$$

Note: $\varnothing=\{\}$, $\{\varnothing\}=\{\{\}\}$, so $\varnothing\neq \{\varnothing\}$.

The empty set is the set that has no elements. It is denoted as $\varnothing$ and it represents $\{\}$. The empty set is the only set whose cardinality is zero.

• $\varnothing=\{\}\neq\{\{\}\}=\{\varnothing\}$

• the empty set is regarded as an existing object.

• empty set is a subset of any set

• empty set is a proper subset of any non-empty set