Lesson 2 of 11 · 7 min
The empty set
NCERT §1.3
The sheet also has a column for kabaddi. Nobody signed it. Is the kabaddi list still a set, and if so, how many elements does it have?
The lesson in notes
In short
A set with no element at all is called the empty set, the null set or the void set, and is written φ or { }.
Sets often turn out empty because the condition cannot be met: no natural number lies strictly between 1 and 2, no rational number squares to 2, no even prime is greater than 2, and no odd number squares to 4.
φ is not {0} and not {φ}: {0} has one element (the number zero) and {φ} has one element (the empty set itself), so neither of them is empty.
Whether a set is empty can depend on where x is allowed to come from: {x : x² = 2} is empty if x must be rational, but it is {√2, −√2} if x may be any real number.
The empty set is counted as a finite set, with n(φ) = 0.