Category Archives: Set Theory and Logic

1.4 Exercise 5

Posted on October 10, 2009 by Carlos

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 →

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1.4 Exercise 4

Posted on September 10, 2009 by Carlos

“(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 →

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1.4 Exercise 3

Posted on July 31, 2009 by Carlos

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 →

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1.4 Exercise 2

Posted on July 25, 2009 by Carlos

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 →

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1.4 Exercise 1

Posted on June 19, 2009 by Carlos

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 →

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1.3 Exercise 15

Posted on June 18, 2009 by Carlos

Yes, after months, I’m back ;-).  This was an interesting problem to me because it explores a bit more deeply the concept of the LUBP. “Assume that the real line has the least upper bound property. (a) Show that the … Continue reading →

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1.3 Exercise 14

Posted on March 11, 2009 by Carlos

Rather than merely repeating the proof of 1.3.13 to show the converse, we can impose the “opposite” or symmetric order relation on the set to show that if it has the GLBP then it has the LUBP too. “If is … Continue reading →

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1.3 Exercise 13

Posted on March 2, 2009 by Carlos

This theorem (and its converse) is needed everywhere in analysis and topology, and it is very important. “Prove the following:  Theorem.  If an ordered set has the least upper bound property, then it has the greatest lower bound property.” (Taken … Continue reading →

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1.3 Exercise 12

Posted on February 27, 2009 by Carlos

This problem, although cool because it exposes us to out-of-the-ordinary order relations on the Cartesian plane, was a little long, I felt.  Nevertheless, it shows that we can create order relations with smallest elements, with elements with no immediate predecessors, … Continue reading →

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1.3 Exercise 11

Posted on February 20, 2009 by Carlos

“Show that an element in an ordered set has at most one immediate successor and at most one immediate predecessor.  Show that a subset of an ordered set has at most one smallest element and at most one largest element.” … Continue reading →

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