Can Floor be equal to Ceiling? - 2

Algebra Level 3

$\Large 1^{\lfloor x \rfloor } = 1^{\lceil x \rceil }$

The set containing all values of $x$ that satisfy the above equation is __________.

Notation: $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions respectively.

Countably Infinite Uncountably infinite Empty Non-empty finite

1 solution

Pranshu Gaba
Mar 9, 2016

For any real number $x$ , both $1^{\lfloor x \rfloor }$ and $1^{\lceil x \rceil }$ are both equal to 1 since one raised to any real number is one itself. Therefore every real number $x$ will satisfy the given equation.

Thus the set containing all values of $x$ that satisfy the given equation is the set of all real numbers, $\mathbb{R}$ . It is an uncountably infinite set. $_\square$

Can Floor be equal to Ceiling? - 2

1 solution

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