September 2010 S M T W T F S « Aug 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Categories
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Category Archives: The Integers and the Real Numbers
1.4 Exercise 5
I had often wondered how to go about proving closure of addition and multiplication of the integers. After this problem I wondered no more! It’s pretty neat that we can show (by using induction) that by adding any two integers … Continue reading
1.4 Exercise 4
“(a) Prove by induction that given , every nonempty subset of has a largest element. (b) Explain why you cannot conclude from (a) that every nonempty subset of has a largest element.” (Taken from Topology by James R. Munkres, Second … Continue reading
1.4 Exercise 3
To get acquainted with the set of positive integers and how this set is related to “proving things by induction,” this problem is a great primer! “(a) Show that if is a collection of inductive sets, then the intersection of … Continue reading
1.4 Exercise 2
Another one of those really really long problems, but oh well… after this one the exercises seem more interesting. As before, I’m doing five by five until the end. “Prove the following laws of inequalities for , using axioms (I)-(VI) … Continue reading
1.4 Exercise 1
Omigod, this was a kilometric problem, and kind of boring too. I guess once in a lifetime every mathematician (or schoolboy) should go ahead and get his hands dirty proving identities only using axioms. Here goes, although I’ll complete this … Continue reading