Complete Notes.pdf - CHAPTER 3 THE REAL NUMBERS PRELIMINARIES Absolute Value Def Let x \u2208 R The absolute value of x denoted | x | is |x| = x \u2212x if x

Complete Notes.pdf - CHAPTER 3 THE REAL NUMBERS...

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CHAPTER 3: THE REAL NUM- BERS PRELIMINARIES: Absolute Value Def. Let x R . The absolute value of x , denoted | x | , is: | x | = x, if x 0 - x, if x < 0
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Properties of absolute value:Letx, yRand leta0. Then(a)|x| ≥0.(b)|x| ≤aiff-axa.(c)|xy|=|x| |y|.(d)|x+y| ≤ |x|+|y|.(Triangleinequality)
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Section 3.3. Boundedness and the Completeness Axiom There is a one-to-one correspondence between the set R of real numbers and the points P on the number line. There are two basic types of real numbers: 1. The rational numbers Q , and 2. the irrational numbers R - Q .
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Def. Let S R , S 6
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