1 Interval Notations and Examples

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When we look at the problem facing a household, we often have to restrict the choice set for example to an interval.

1.1 Closed Interval

For example, if \(x\) is hours working, perhaps the household has to work at least \(a\) hours and up to \(b\) hours, so his choice is between \(a\) and \(b\) hours inclusive.

The interval that is inclusive of both endpoints is called a closed interval (note the square brackets):

  • closed interval: \(\left\lbrack a,b\right\rbrack \equiv \lbrace x\in R:a\le x\le b\rbrace\)

1.2 Open Interval

In general, an open interval is defined as (Note here we use parenthesis, not square brackest) :

  • open interval:\(\left(a,b\right)\equiv \lbrace x\in R:a<x<b\rbrace\)

1.3 Half Open and Half Close Interval

We can also hafl half open intervals:

  • half open (half closed) interval: \(\left\lbrack a,b\right)\equiv \lbrace x\in R:a\le x<b\rbrace\)

  • half open (half closed) interval: \(\left(a,b\right\rbrack \equiv \lbrace x\in R:a<x\le b\rbrace\)

1.4 Graph

If you were to graph an interval, you can draw an empty circle at either end of an interval that is open, and a solid circle if it is closed at that end.

close all;
figure();
x = linspace(-1,5);
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