Question

Define an empty set.

Answer 1

Hint: To define this first we have to know what is an empty set and how it is denoted.Empty set is denoted as \[\phi \]

Complete step by step solution:
The empty set is a unique set having no element, its size and cardinality is zero.
The symbol of empty set is $\{ \;\} $ or \[\phi \]
For Example:
A. \[{\text{M = }}\](\[{\text{d: d > 8, d}}\] is the number of days in a week)
Explanation:
Here,\[{\text{d}}\]is greater than\[8\].So, this set (\[{\text{M}}\]) is an empty set because there are only\[7\]days in a week.
B. The set of squares with$5$sides

Explanation:
The figure is shown as a square in which it is clear that the square has four sides. So, it is impossible to have $5$ sides of square as a figure which have five sides that figure is known as pentagon

Properties of empty set:-
1. The empty set is a subset of${\text{A}}$:
\[\forall A:\phi \subseteq A\]
2. The union of${\text{A}}$with the empty set is${\text{A}}$:
\[\forall A:A \cup \phi = A\]
3. The intersection of${\text{A}}$with the empty set is the empty set
\[\forall A:A \cap \phi = \phi \]
4. The Cartesian product of${\text{A}}$and the empty set is the empty set
\[\forall A \times \phi = \phi \]


Note: Take care to see that when giving example of an empty set in rule method,when expanded the set should have no elements in it