1 Real Number and intervals

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1.1 Real Number Line

\({R^1 }\) : can write \(R^1\) or \(R\) (you can add a superscript 1 to emphasize this is first Euclidean space, either notation is fine), is the real number line.

close all;
figure();
x = linspace(-10,10);
line(x,0*ones(size(x)))
set(gca,'ytick',[],'Ycolor','w','box','off')
ylim([-0.1 0.1])
pbaspect([4 1 1])
grid on

1.2 Non-negative numbers

In many economic problems, we have to restrict ourselves to numbers greater or equal to zero.

  • We can not consume from negative numbers of apples

  • We can not produce with negative labor and capital

  • We would be infinitely unhappy (die) if there is zero consumption in a year

We can use the following notation to define the set of non-negative real numbers:

\({R_{\ge 0} }\equiv \lbrace x\in R:x\ge 0\rbrace\), some authors use \({R_+ }\) instead of \({R_{\ge 0} }\)

And use inequality sign to define the set of real numbers greater than zero:

\({R_{>0} }\equiv \lbrace x\in R:x>0\rbrace\), some authors use \({R_{++} }\) instead of \({R_{>0} }\)

close all;
figure();
x = linspace(0,10);
line(x,0*ones(size(x)))
set(gca,'ytick',[],'Ycolor','w','box','off')
ylim([-0.1 0.1])
xlim([-10 10])
pbaspect([4 1 1])
grid on