3al05_2016.pdf

# 3al05_2016.pdf - 1/46 1 Lecture 2 Properties of the Real...

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1/46 1 Lecture 2 Properties of the Real Numbers 2 Lecture 3 Properties of the Real Numbers II 3 Lecture 4 Properties of the Real Numbers III 4 Lecture 5 Properties of the Real Numbers IV Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 2 Properties of the Real Numbers 2/46 Mathematics and Statistics Z M d ω = Z M ω Mathematics 3A03 Real Analysis I Instructor: David Earn Lecture 2 Properties of the Real Numbers Friday 9 September 2016 Instructor: David Earn Mathematics 3A03 Real Analysis I
Lecture 2 Properties of the Real Numbers 3/46 Where to find course information The course wiki: Click on Lectures Click on Course information to download pdf file. Check the course wiki regularly! If preliminary lecture slides are ready the night before, they will be posted. However, you must download the lecture slides posted AFTER class in order to obtain final versions. Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 2 Properties of the Real Numbers 4/46 What we did last class The “Reals” ( R ) are all the numbers that are needed to fill in the “number line” (so it has no “gaps” or “holes”). The rationals ( Q ) have “holes”, e.g., 2. Numbers can be constructed using sets. We will discuss this informally . A more formal approach is taken in Math 4L03 (Mathematical Logic) or in this online e-book . The naturals ( N = { 1 , 2 , 3 , . . . } ) can be constructed from : 0 = , 1 = { 0 } , 2 = { 0 , 1 } , . . . , n + 1 = n ∪ { n } . The integers ( Z ), and operations on them (+ , - , · ), can also be constructed from sets and set operations (but we deferred that for later). Given N and Z , we can construct Q . . . Instructor: David Earn Mathematics 3A03 Real Analysis I
Lecture 2 Properties of the Real Numbers 5/46 Informal introduction to construction of numbers ( Q ) Rationals: Idea: Q = Z × N Use notation a b for rational number ( a , b ) Q . Define equivalence of rational numbers: a b = c d ⇐⇒ a · d = b · c Define order for rational numbers: a b c d ⇐⇒ a · d b · c Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 2 Properties of the Real Numbers 6/46 Informal introduction to construction of numbers ( Q ) Rationals, continued: Define operations on rational numbers: a b + c d def = ad + bc bd a b · c d def = a · c b · d Constructed in this way (ultimately from the empty set), Q satisfies all the standard properties we associate with the rational numbers. Instructor: David Earn Mathematics 3A03 Real Analysis I
Lecture 2 Properties of the Real Numbers 7/46 Properties of the rational numbers ( Q ) Addition: A1 Closed and commutative under addition. For any x , y Q there is a number x + y Q and x + y = y + x . A2 Associative under addition. For any x , y , z Q the identity ( x + y ) + z = x + ( y + z ) is true. A3 Existence and uniqueness of additive identity. There is a unique number 0 Q such that, for all x Q , x + 0 = 0 + x = x .

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