Empty Set/Examples/Real Roots of x^2 + 1

Example of Empty Set

The set:

$S = \set {x \in \R: x^2 + 1 = 0}$

is an instance of a specification of the empty set.

Proof

From Existence of Real Polynomial with no Real Root, there exists no $x \in \R$ such that $x^2 + 1 = 0$.

Hence the result.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.1$: Set Notation