Real Analysis, Lecture 1: Constructing the Rational Numbers

Real Analysis, Spring 2010, Harvey Mudd College, Professor Francis Su. Get notes and study tools at http://gosuapm.com/

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Education

Tags:

• harvey mudd
• hmc
• su
• analysis
• math

[WorldCollections]

Now here's a professor that knows how to teach! ...he does know the golden rule: if you can make it fun...why not? Some other guys just seem like they are teaching a wall...instead of groups of students; that, they can do at their homes! Nice class, even though I would've love to some more light...but I guess its ok...you know, its free!

[Nisshoku1729]

yo dawg, I heard you like sets.

[miami360x]

If you're taking real analysis during your freshman or sophmore year, you are pretty badass

[ugoolm]

malose lol



[stijnisgoed]

@KillerNedDude f(x)=1 for all x in R satisfies the condition for a function: for a given input x, there is a unique output: 1. It does not have to be the other way around: a function has to turn an input into a unique output, not the other way around (if it does, it's a bijection). In a normal function plot, this means: on every vertical line (so a specific x), there can be at most one point y=f(x) on the plot (which f(x)=1 for all x satisfies).

[KillerNedDude]

@stijnisgoed Cheers that makes sense, but what about say f(x) = 1 (for all x in R), is this not a function? Also are functions such as f(x) = { 0 if x is even, 1 if x is odd} (for all x in Z) not valid functions? (I'm guessing I probably have to study the course more to find out).

[stijnisgoed]

@stijnisgoed Correction: I meant 2 values for every input (accept zero).

[stijnisgoed]

@KillerNedDude That's a good question. Strictly speaking, you should also give the domain and codomain when giving a function. The arcsin function is normally defined to be from the input interval [-1,1] to the output interval [-pi/2,pi/2], in which case it has a unique output for every input (and is even a bijection). It's like the square root function, which goes from all non-negative real numbers to just the non-negative real numbers, in stead of giving 2 values for every function.

[KillerNedDude]

@stijnisgoed but then what is unique about the outputs of say the inverse sine function which are the same for many different inputs i.e. asin(0) = 0, pi or npi (for all n in Z).

[stijnisgoed]

@KillerNedDude He says that, given a specific input, a function has a unique output, which is correct. A function is surjective if every member of its codomain is the function value of at least one input, and a function is bijective if every member of its codomain can be traced back to exactly one input. So f(x)=x^2 (R to R) is a function, but not bijective because (2)^2=(-2)^2=4, and 4 cannot be traced back to one unique input. It's not even surjective, because no x satisfies f(x)= -1.

[KillerNedDude]

Not sure about his description of functions in this lecture, he say's that a function should have a unique output, surley this is only the case with a bijective function.