Set Theory Proof of $m^n < 2^{mn}$ for $m,n \in Z^{+}$

$n^m<2^{mn}$ for $m,n \in Z^{+}$

I chanced upon a rather interesting proof of this while studying sets, relations and functions in math class. Try to see if you can find it!

To get you started, consider two non-empty sets $A$ and $B$ containing $m$ and $n$ elements respectively. I'll post the solution in a few days...

SOLUTION

Consider two non-empty sets $A$ and $B$ containing $m$ and $n$ elements respectively.
The CARTESIAN PRODUCT of $A$ and $B$ , given by $A \times B = \{(a,b):a \in A, b \in B\}$ , is the set that contains all possible ordered pairs of the form $(a,b)$ . Obviously, this set contains $mn$ elements.
A RELATION between two sets $A$ and $B$ is defined as a subset of their Cartesian product. Then, how many relations are possible between $A$ and $B$ ? This is the number of possible subsets of the Cartesian product, or, in other words, the number of elements in the POWER SET of $A \times B$ . How do we find the number of possible subsets of $A \times B$ ? To choose a subset from a set, what do we do? We either choose an element to be part of the subset or not, i.e. we make a choice between 2 alternatives. Since there are $mn$ elements in the set, and we need to make a choice for each element, the number of subsets possible is given by $\boxed{2^{mn}}$ .
A FUNCTION between two sets $A$ and $B$ is a subset of their Cartesian product too (just like a relation) but with an additional constraint: all elements of $A$ should have exactly one image in $B$ . Then, how many functions are possible between $A$ and $B$ ? Since each element in $A$ has $n$ possible choices of an image in $B$ and there are $m$ elements in $A$ , there are a total of $\boxed{n^{m}}$ possible functions from $A$ to $B$ .
The final step is just logic: if a function is a relation with a constraint, there have to be fewer possible functions than relations.

Hence, $n^m < 2^{mn}$ .

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Sort by:

Top Newest

${ m }^{ n }\quad ?{ \quad 2 }^{ mn }\\ { m }^{ n }\quad ?\quad { ({ 2 }^{ m }) }^{ n }\\ m\quad <\quad { 2 }^{ m }\\ { m }^{ n }\quad <{ \quad 2 }^{ mn }$

*I use the ? to show that the relation is not yet known

Niven Achenjang - 6 years ago

Well, of course you're right algebraically, but that's not the curious proof I saw. I'll post the solution soon, maybe in two more days... :)

Raj Magesh - 6 years ago

Subbed for any good solutions other than simple inequality. =3=

Samuraiwarm Tsunayoshi - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Set Theory Proof of mn<2mnm^n < 2^{mn}mn<2mn for m,n∈Z+m,n \in Z^{+}m,n∈Z+

Comments