INTERVALS

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(1) Closed Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b}\) , then \(\left[ \text{a,}\,\text{b} \right]=\left\{ x:x\in \text{R},\,a\le x\le b \right\}\)

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(2) Open Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b,}\) then \(\left] a,b \right[\) or \(\left( a,\,\,b \right)=\left\{ x:x\in \text{R},a<x<b \right\}\)

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(3) Closed-open Interval: Let a and b are two given real numbers such that \(a<b\), then \(\displaystyle \left[ a,b \right[\,\,\text{or}\,\,\left[ a,\,\,b \right)=\{x:x\in \text{R},a\le x<b\}\)

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(4) Open-closed Interval: Let a and b are two given real numbers such that \(a<b,\) then \(\left] a,b \right]\,\,\,or\,\,\left( a,\,b \right]\,\,=\,\left\{ x:\,x\in \,\text{R,}\,a<x\le b \right\}\)

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  •  Real number, R represented as \(\left( -\infty ,\,\infty  \right)\)
  • \({{\text{R}}^{\text{+}}}\) can be represented as \([0,\,\,\infty )\)
  • \({{\text{R}}^{-}}\)can be represented as \([-\infty ,\,0)\)
  • We can’t represent Natural numbers, Integers, whole numbers in intervals.

Examples:

\(\displaystyle \left\{ x:x\in \text{R},-2<x\le 3 \right\}=\left( -2,3 \right]\)

\(\begin{array}{l}\left\{ x:x\in \text{R},-2\le x\le 1\, \right\}=\left[ -2,1 \right]\\\left\{ x:x\in R,-1<x<1 \right\}\,\,\,=\left( -1,\,\,1 \right)\\\left\{ x:x\in \text{R,}\,\text{0}\,\le \,x<5 \right\}\,\,\,\,\,\,=\left[ 0,5 \right)\end{array}\)

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