Can Floor be equal to Ceiling? - 1

Algebra Level 3

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

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.

Uncountably infinite Countably infinite Empty Non-empty finite

2 solutions

Pranshu Gaba
Mar 9, 2016

We can take the natural logarithms on both sides. We get

$\lfloor x \rfloor = \lceil x \rceil$

This equation is true if and only if $x$ is an integer. If $x$ is not an integer, then $\lfloor x \rfloor = \lceil x \rceil -1$ , and the above equation becomes false. Thus the set containing all values of $x$ satisfying is the set of all integers $\mathbb{Z}$ . It is a countably infinite set . $_\square$

Sir, what is the difference between "countably infinite" and "uncountably infinite"? How can infinity be counted? Thanks in advance!

Vinayak Srivastava - 11 months, 2 weeks ago

You can read the Brilliant wiki on cardinality . A set $X$ is countably infinite is there exists a bijection between $X$ and $\mathbb{N}$ , the positive integers. Examples of a countably infinite set include the set of integers $\mathbb{Z}$ , and the set of all rational numbers $\mathbb{Q}$ .

An uncountably infinite set is a set $X$ such that there is no bijection from $X$ to $\mathbb{N}$ . Uncountably infinite sets are much bigger than countably infinite sets. Examples include all real numbers $\mathbb{R}$ , all reals between 0 and 1 $[0, 1]$ , and the set of all complex numbers $\mathbb{C}$ .

Pranshu Gaba - 11 months ago

Ok thank you!

Vinayak Srivastava - 11 months ago

Pulkit Gupta
Mar 9, 2016

The set of integers, Z , satisfy the given equation. And Z is countably infinite .

Can Floor be equal to Ceiling? - 1

2 solutions

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