Few basic properties of inequalities

Transitivity: 

For a, b, c ∈ ℝ  we have that   a≤b   and   b≤c   implies   a≤c

Respect for multiplication: 

For a, b, c ∈ ℝ we have that  a≤b   and   0≤c   implies   a c≤b c

Respect for the reciprocal: 

For  a ∈ ℝ we have that  0<a   implies  0<(1/a)

Review of functions

The sign function

sign  x>0  then Sign[x] = 1, x =0 then Sign[x] = Sign[0] = 0, if x < 0 then Sign[x] = -1

Plot[Sign[x],{x,-3,3},PlotStyle->{Black},Epilog->{{PointSize[0.015],Point[{0,0}]},{PointSize[0.015],Point[{0,1}],PointSize[0.01],White,Point[{0,1}]},{PointSize[0.015],Point[{0,-1}],PointSize[0.01],White,Point[{0,-1}]}}]

The unit step function

unit step function

Plot[UnitStep[x],{x,-3,3}]

UnitStep[0]

Plot[UnitStep[x],{x,-3,3},PlotStyle->{Black},Epilog->{{PointSize[0.015],Point[{0,1}]},{PointSize[0.015],Point[{0,0}],PointSize[0.01],White,Point[{0,0}]}}]

The formal definition of the unit step function is

x∈ℝ if x<0 then us(x) = 0, if x≥0 then us(x) = 1

The floor function

Plot[{x,Floor[x]},{x,-8,8},PlotStyle->{{RGBColor[1,0.5,0]},{Black}},Epilog->{{PointSize[0.015],Black,Table[Point[{k,k}],{k,-10,10}]},{PointSize[0.015],Table[Point[{k+1,k}],{k,-10,10}],PointSize[0.01],White,Table[Point[{k+1,k}],{k,-10,10}]}},AspectRatio->Automatic,PlotRange->{{-4.5,4.5},{-4.5,4.5}},Ticks->{Range[-5,5],Range[-5,5]},ImageSize->300]

Formal definition of the floor function! 

For  x∈ ℝ  we define:  Floor[x]=⌊x⌋= max{k∈ℤ:k≤x}

The ceiling function

Plot[{x,Ceiling[x]},{x,-8,8},PlotStyle->{{RGBColor[1,0.5,0]},{Black}},Epilog->{{PointSize[0.015],Black,Table[Point[{k,k+1}],{k,-10,10}]},{PointSize[0.015],Table[Point[{k,k}],{k,-10,10}],PointSize[0.01],White,Table[Point[{k,k}],{k,-10,10}]}},AspectRatio->Automatic,PlotRange->{{-4.5,4.5},{-4.5,4.5}},Ticks->{Range[-5,5],Range[-5,5]}]

Formal definition of the ceiling function! 

For  x∈ ℝ  we define:  Ceiling[x]=⌈x⌉= min{k∈ℤ:k≥x}

Ceiling[Pi]

The absolute value function

abs

Plot[{Abs[x]},{x,-4,4},AspectRatio->Automatic,PlotRange->{{-4,4},{-1,4}},Ticks->{Range[-5,5],Range[-5,5]}]

Think of the real number line as a special highway. How far is  π  from e?   π-e

Think of the real number line as a special highway. How far is  (π^e)  from (e^π)?  |(π^e)-(e^π)|

The definition of the absolute value function. Let x∈ℝ. Then if x<0 then abs(x)=-x, if x≥0 then abs(x) = x.

Exercise: Prove abs(x) = x sign(x) for all x∈ℝ.

Triangle inequality