Positive and negative parts

Positive and Negative Parts of Template:Math

In mathematics, the positive part of a real or extended real-valued function is defined by the formula $$f^{+}(x)=\max (f(x),0)=\{\begin{array}{cc}f(x)& \text{ if }f(x)>0\\ 0& \text{ otherwise.}\end{array}$$

Intuitively, the graph of $f^{+}$ is obtained by taking the graph of $f$, chopping off the part under the Template:Math-axis, and letting $f^{+}$ take the value zero there.

Similarly, the negative part of Template:Math is defined as $$f^{-}(x)=\max (-f(x),0)=-\min (f(x),0)=\{\begin{array}{cc}-f(x)& \text{ if }f(x)<0\\ 0& \text{ otherwise}\end{array}$$

Note that both Template:Math and Template:Math are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part).

The function Template:Math can be expressed in terms of Template:Math and Template:Math as $$f=f^{+}-f^{-}.$$

Also note that $$|f|=f^{+}+f^{-}.$$

Using these two equations one may express the positive and negative parts as $$\begin{array}{cc}{f}^{+}& =\frac{|f|+f}{2}\\ f^{-}& =\frac{|f|-f}{2}.\end{array}$$

Another representation, using the Iverson bracket is $$\begin{array}{cc}{f}^{+}& =[f>0]f\\ f^{-}& =-[f<0]f.\end{array}$$

One may define the positive and negative part of any function with values in a linearly ordered group.

The unit ramp function is the positive part of the identity function.

Measure-theoretic properties

Given a measurable space Template:Math, an extended real-valued function Template:Math is measurable if and only if its positive and negative parts are. Therefore, if such a function Template:Math is measurable, so is its absolute value Template:Math, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking Template:Math as $$f=1_V-\frac{1}{2},$$ where Template:Math is a Vitali set, it is clear that Template:Math is not measurable, but its absolute value is, being a constant function.

The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.

See also

• Rectifier (neural networks)
• Even and odd functions
• Real and imaginary parts

References

• Template:Cite book
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External links

• Positive part on MathWorld