Empty Set Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium

Let

A = \emptyset

. Find: (a)

A \cup B

for any set

B

, (b)

A \cap B

for any set

B

, (c)

\mathcal{P}(A)

(the power set of

A

Solution

1
(a) $\emptyset \cup B = B$ . Adding no elements to $B$ leaves $B$ unchanged.
2
(b) $\emptyset \cap B = \emptyset$ . There are no elements in $\emptyset$ , so the intersection is empty.
3
(c) $\mathcal{P}(\emptyset) = \{\emptyset\}$ . The only subset of $\emptyset$ is $\emptyset$ itself, so the power set has one element.

Answer

(a)\;B,\quad (b)\;\emptyset,\quad (c)\;\{\emptyset\}

The empty set is an identity element for union and an absorbing element for intersection. Its power set

\{\emptyset\}

has exactly one element, so

|\mathcal{P}(\emptyset)| = 1 = 2^0

About Empty Set

The empty set, denoted $\emptyset$ or $\{\}$ , is the unique set that contains no elements at all. It is a subset of every set because the statement 'every element of $\emptyset$ belongs to $A$ ' is vacuously true — there are no elements to contradict it.

Learn more about Empty Set →

More Empty Set Examples

Example 1 easy

Determine whether each set is empty: (a) [formula], (b) [formula], (c) [formula].

Example 2 medium

Prove that the empty set [formula] is a subset of every set [formula].

Example 3 easy

Decide which are true: (a) [formula], (b) [formula], (c) [formula].

← All Empty Set Examples Empty Set Concept

Related Concepts

Set Cardinality