<p>An example of a locally monotone function is $g(x,y) = x \cdot (1-y)$. In this example $g$ is monotone increasing in $x$ and monotone decreasing in $y$.</p>

<p>An example of a function that is not locally monotone is $h(x,y) = x \oplus y$.
Let's see why it is not locally monotone. Look at the $x$ variable. (1) It is neither monotone increasing in $x$ since $h(0,1) &gt; h(1,1)$, (2) nor is it monotone decreasing in $x$ since $h(0,0) &lt; h(1,0)$.</p>

<p>Locally monotone functions are called also <a href="http://en.wikipedia.org/wiki/Unate_function" rel="nofollow">unate functions</a>.</p>

CC BY-SA 3.0
2013-05-13T06:43:47.800Z
at the end it will be $h(0,0) < h(1,0)$
Kumar
CC BY-SA 3.0
2013-05-13T06:57:11.977Z
right, sorry. i just updated the answer
Igor Shinkar
CC BY-SA 3.0
2013-05-13T06:03:08.523Z
2013-05-13T07:03:15.410Z
2013-05-13T07:03:15.410Z
Igor Shinkar
<p>I am unable to understand the definition of locally monotone Boolean function which is defined in Gotsman and Linial, "<a href="http://www.cs.huji.ac.il/~nati/PAPERS/gotsman_linial.pdf" rel="nofollow">Spectral Properties of Threshold Functions</a>", 1994, p. 40:</p>

<blockquote>
  <p>A function $f$ is called <em>locally monotone</em> if it is increasing or decreasing in each variable.</p>
</blockquote>

<p>Can somebody explain the definition with two examples (a function which is locally monotone and a function which is not)?</p>

CC BY-SA 3.0
2013-05-13T06:29:11.123Z
spectral properties of threshold functions by Gotsman and Linial, 1994.
Kumar
<cc.complexity-theory><boolean-functions><combinatorics>
Locally monotone Boolean function
Kumar
# Locally monotone Boolean function

(Asked by: username_0 on 2013-05-13)

I am unable to understand the definition of locally monotone Boolean function which is defined in Gotsman and Linial, "[Spectral Properties of Threshold Functions](http://www.cs.huji.ac.il/~nati/PAPERS/gotsman_linial.pdf)", 1994, p. 40:

> A function $f$ is called *locally monotone* if it is increasing or decreasing in each variable.

Can somebody explain the definition with two examples (a function which is locally monotone and a function which is not)?

**Tags:** cc.complexity-theory, boolean-functions, combinatorics
> **Comments:**
> **username_0**: spectral properties of threshold functions by Gotsman and Linial, 1994.  

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## 1 answers

### Answer by username_1 on 2013-05-13. Score: 5 | Accepted answer

An example of a locally monotone function is $g(x,y) = x \cdot (1-y)$. In this example $g$ is monotone increasing in $x$ and monotone decreasing in $y$.

An example of a function that is not locally monotone is $h(x,y) = x \oplus y$.
Let's see why it is not locally monotone. Look at the $x$ variable. (1) It is neither monotone increasing in $x$ since $h(0,1) > h(1,1)$, (2) nor is it monotone decreasing in $x$ since $h(0,0) < h(1,0)$.

Locally monotone functions are called also [unate functions](http://en.wikipedia.org/wiki/Unate_function).

> **Comments:**
> **username_0**: at the end it will be $h(0,0) < h(1,0)$  
> **username_1**: right, sorry. i just updated the answer  

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