Editing Infimum and supremum (section)

===Arithmetic operations on sets===

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. 
Throughout, <math>A, B \subseteq \R</math> are sets of real numbers. 

'''Sum of sets'''

The [[Minkowski sum]] of two sets <math>A</math> and <math>B</math> of real numbers is the set 
<math display=block>A + B ~:=~ \{a + b : a \in A, b \in B\}</math> 
consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfy, if <math>A \ne \varnothing \ne B</math>
<math display=block>\inf (A + B) = (\inf A) + (\inf B)</math> 
and 
<math display=block>\sup (A + B) = (\sup A) + (\sup B).</math>

'''Product of sets'''

The multiplication of two sets <math>A</math> and <math>B</math> of real numbers is defined similarly to their Minkowski sum:
<math display=block>A \cdot B ~:=~ \{a \cdot b : a \in A, b \in B\}.</math>

If <math>A</math> and <math>B</math> are nonempty sets of positive real numbers then <math>\inf (A \cdot B) = (\inf A) \cdot (\inf B)</math> and similarly for suprema <math>\sup (A \cdot B) = (\sup A) \cdot (\sup B).</math><ref name="zakon">{{cite book|title=Mathematical Analysis I|first=Elias|last=Zakon|pages=39–42|publisher=Trillia Group|date=2004|url=http://www.trillia.com/zakon-analysisI.html}}</ref>

'''Scalar product of a set'''

The product of a real number <math>r</math> and a set <math>B</math> of real numbers is the set
<math display=block>r B ~:=~ \{r \cdot b : b \in B\}.</math>

If <math>r > 0</math> then 
<math display=block>\inf (r \cdot A) = r (\inf A) \quad \text{ and } \quad \sup (r \cdot A) = r (\sup A),</math>
while if <math>r < 0</math> then 
<math display=block>\inf (r \cdot A) = r (\sup A) \quad \text{ and } \quad \sup (r \cdot A) = r (\inf A).</math>
In the case <math>r = 0</math>,
one has, if <math>A \ne \varnothing</math>
<math display=block>
  \inf (0 \cdot A) = 0 \quad \text{ and } \quad  \sup (0 \cdot A) = 0 
</math> 
Using <math>r = -1</math> and the notation <math display=inline>-A := (-1) A = \{- a : a \in A\},</math> it follows that, 
<math display=block>\inf (- A) = - \sup A \quad \text{ and } \quad \sup (- A) = - \inf A.</math>

'''Multiplicative inverse of a set'''

For any set <math>S</math> that does not contain <math>0,</math> let
<math display=block>\frac{1}{S} ~:=\; \left\{\tfrac{1}{s} : s \in S\right\}.</math>

If <math>S \subseteq (0, \infty)</math> is non-empty then 
<math display=block>\frac{1}{\sup_{} S} ~=~ \inf_{} \frac{1}{S}</math> 
where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used.<ref group="note" name="DivisionByInfinityOr0">The definition <math>\tfrac{1}{\infty} := 0</math> is commonly used with the [[extended real number]]s; in fact, with this definition the equality <math>\tfrac{1}{\sup_{} S} = \inf_{} \tfrac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\tfrac{1}{0}</math> is usually left undefined, which is why the equality <math>\tfrac{1}{\inf_{} S} = \sup_{} \tfrac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math></ref> 
This equality may alternatively be written as 
<math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \tfrac{1}{s}.</math> 
Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \tfrac{1}{S} = \infty,</math> where if<ref group=note name="DivisionByInfinityOr0" /> <math>\inf_{} S > 0,</math> then <math>\tfrac{1}{\inf_{} S} = \sup_{} \tfrac{1}{S}.</math>