Sets

• Cantor, Peano, Natural Numbers, and Infinity [Bennett, 03/19/1998]
  A conversation on transfinite numbers and contradictions the questioner believes exist in Cantor's paper introducing the diagonal method.

• Cardinal and Ordinal Numbers [Aitchison, 01/08/1997]
  How can 3 be both a cardinal and ordinal number at the same time?

• Defining Kinds of Numbers [Bunting, 03/21/1997]
  Could you please define: perfect numbers, deficient numbers, square numbers, abundant numbers, amicable numbers, and triangular numbers?

• Difference Between Zero and Nothing [Cockburn, 12/12/1996]
  What is the difference between zero and nothing?

• Infinite Sets [Lee, 07/17/1997]
  How do you prove that there are more rational numbers than negative integers? How can you tell if an infinite set is countable or uncountable?

• Intersection of Sets [Kristi, 10/02/2000]
  I do not understand intersection of sets. Can you give me an example?

• Line Segments and Size of Infinites [Allen, 03/19/1997]
  Divide a line segment into three parts, one half and one a quarter the length of the line segment. Choose a point at random along this line segment. What is that probability that this point lands in the 1/2 segment...?

• Lines, Points, and Infinities [Graham, 09/01/2001]
  What is the cardinality of the set of real numbers between 0 and 1? Is this cardinality less than, greater than, or equal to the cardinality of real numbers between 0 and 2?

• One-to-One Correspondence and Transfinite Numbers [Huang, 12/10/1999]
  Can you explain what Cantor meant by one-to-one correspondence, and transfinite numbers?

• Rational and Irrational Numbers [Merrill, 11/12/1997]
  Which set is bigger, the set of rational or irrational numbers?

• Rational Numbers [Oliver, 11/24/1997]
  Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?

• Set, Subset, Element [Lau, 3/10/1997]
  Please define: set, subset, member, element, intersection, union.

• Set Theory and Orders of infinity [Sergio, 04/08/1997]
  Given a lists of sets, such as all real numbers between 0 and 1, the integers, the odd integers, etc. how do I compare their size? And what does this have to do with Cantor's set theory?

• Sets and Subsets [Li, 1/23/1995]
  My teacher said that integers are a subset of reals, and whole numbers are a subset of integers, and counting numbers are a subset of whole numbers, and so on and so forth. What does that mean?

• Sets: Unions and Intersections [Lawless, 12/17/1997]
  I want to know about complements, union, intersection, and sets of numbers.

• Tree Diagram for Math Numbers [Hallock, 10/05/1997]
  My daughter is doing a tree diagram using terms related to math "numbers." Could you please explain in lay terms what surds are?

• Unions and Intersections [Warren, 2/9/1995]
  In my text, there are these upside-down horseshoe looking things, and there is no explanation of what they are or why they exist...

• What are Sets and Subsets? [Andrea, 09/06/2001]
  Can you please give me examples of sets and subsets?

• What is a Set? [Cifuentes, 04/04/1997]
  What is the correct term to refer to groups of objects like 3 cars, 7 pencils, or 5 apples?

• Infinity, Zero [Charny, 1/4/1995]
  You can't divide by zero, but no one can actually prove WHY. . . I wanted to see a real proof first. . . We learned in trig that you can't raise zero to the zero-th power because zero would equal one, obviously. I realize infinity is not so much a number as an endless amount, but if there are an infinite number of numbers between 1 and 2, and an infinite number of numbers between 1 and 50, wouldn't the second infinity be bigger than the first?

• Empty Set Classification [Lowe, 2/20/1995]
  We would like to know if you would classify an empty set as being finite, infinite, or with another classification.

• Equivalence Relations on Sets [iLbt, 2/3/1996]
  Please tell me how many equivalence relations can be defined on the set S = [a,b,c].

• Infinite and Transfinite Numbers [Adams, 5/28/1996]
  Can anyone explain to me, in a simple way, what transfinite numbers are and how they're different from infinite numbers?

• Null Sets [Jenkins, 8/23/1996]
  Is the null set a proper subset of every set?

• How Many are in the Group? [Benedict, 10/17/1996]
  Everyone in the group had been to at least one of the parks...

• One to One Correspondence Between Sets [Nicholas, 04/04/1997]
  What is the one-to-one correspondence between the set of natural numbers and the set "S" = {1/3, 1/6, 1/12, 1/24, 1/48, ...}?

• Borel fields [Mooney, 08/10/1997]
  From the definition given in my book for an algebra, I don't understand why EVERY algebra would not be a Borel field.

• Disjoint Subsets, Complement, Cardinality [Wang, 09/03/1997]
  What are disjoint subsets? What are "complement" and "cardinality"?

• Interval as Intersection of Sets [Clase, 09/04/1997]
  - 4 is less than or equal to y is greater than or equal to 6. Describe the interval as the intersection of two other sets.

• Infinite Proper Subset of an Infinite Set [Cotten, 09/22/1997]
  Given set A = {1,2,3,...} and set B = {10,20,30,...}, is B a proper subset of A?

• Infinite Sets [Gibbs, 09/24/1997]
  In my algebra class we have been debating whether the integers or the whole numbers contain more elements...

• What is a Loop? [Gardner, 10/06/1997]
  Further, exactly what is _an_ algebra? Thirdly, what is a Lie group?

• Divisibility of Zero Theory [ONeal, 10/06/1997]
  A student claims that he has heard of divisibility OF zero theory... can you fill me in on this concept?

• Abundant and Deficient Numbers [Byers, 10/14/1997]
  What are abundant and deficient numbers, and what are they used for?

• Raising Sqrt(2) [Benno, 10/14/1997]
  What is the value of x in the following equation: x^x^x^x^x ... = 2 ?

• Induction Problem [Wesolowski, 10/14/1997]
  Use math induction to prove that (1+2+3+...n)^2 = 1^3+2^3+3^3...n^3.

• Induction Proof [Bryce, 10/15/1997]
  Examine the values of the expression 1/1.2 + 1/2.3 + .... + 1/n(n+1), find the formula, and prove the result.

• Transfinite Arithmetic [Simakovsky, 10/28/1997]
  What is transfinite arithmetic? I pretty much know what it means, but I am having trouble applying it to aleph-null.

• Prove That a Set Is Uncountably Infinite [Ramos, 10/31/1997]
  How can one prove that the set [0,1]x[0,1] is uncountably infinite?

• Transfinite Numbers [McSwain, 11/07/1997]
  I know that Georg Cantor discovered transfinite numbers, but what are they?

• Aleph Null [Pastor, 01/22/1998]
  What does aleph null represent?

• Positive Numbers Less Than -3 [Heather, 01/30/1998]
  Write the set C of positive multiples of 3 less than -3 using roster notation and set-builder notation. How can a positive number be less than -3?

• Density of Rational Numbers [McKenzie, 01/31/1998]
  I am wondering what the following statement means: "The set of rational numbers is dense."

• Sets Containing an Infinite Number of Members [Kate, 02/03/1998]
  Which of these two sets is larger - the set of all rational numbers, or the set of all irrational numbers?

• Overlapping Sets [Puri, 02/10/1998]
  We are given 87 tibbs. All 34 gibbs and 49 pibbs are tibbs. If exactly 9 tibbs are gibbs and pibbs, then how many tibbs are neither pibbs nor gibbs?

• Venn Diagrams [Chris, 02/26/1998]
  What is a Venn Diagram? What is its use, definition, and what does it look like?

• Polynomial Degrees and Definition of a Field [Metelli, 03/02/1998]
  The degree of polynomials added together, and definition of a field.

• Russell's Infinite Set Paradox [McAllister, 03/25/1998]
  Given the set (S) of all sets that do not contain themselves, does S contain itself?

• Closure and the Reals [Zobell, 03/26/1998]
  Under what set of operations are the positive real integers closed?

• Proving a Set is Closed [Okie, 04/15/1998]
  Let f be a continuous real valued function on Rn, and let S = {x in Rn : f(x) = 1}. Prove that S is a closed subset of Rn.

• Connected Sets in Topology [Flowers, 04/22/1998]
  Exploring connected sets with examples in Euclidean space.

• Finite and Infinite Ordinals [DJ, 06/19/1998]
  I know that you can add infinity to infinity to get infinity, but what happens when you multiply infinity times infinity?

• Sets: Cardinality and Functions [Marco, 06/29/1998]
  What does "{1,0}^{a,b}" mean?

• Number Systems: Two Points of View [Liz, 06/30/1998]
  What are the number systems?

• Subsets and Proper Subsets [Dwelly, 07/10/1998]
  What is the difference between a subset and a proper subset?

• John Venn and Venn Diagrams [Potemra, 09/04/1998]
  Can you give me some information on John Venn and the origin of Venn diagrams?

• The Real Number System in a Venn Diagram [Amy, 09/08/1998]
  How do you construct a Venn diagram that illustrates the real number system?

• Definitions and Mappings of Sets [Wang, 09/22/1998]
  I am studying mathematical analysis out of Rudin and have some questions...

• Open Balls and Open Sets [Robert, 10/04/1998]
  Can you explain the concepts of open balls and open sets? What about in the plane (2D) and in space (3D)?

• Set Equality [Lior, 10/12/1998]
  Can you help me show that (A-B)-C = (A-C)-(B-C), where A, B, and C are sets?

• Types of Numbers [Kustera, 10/30/1998]
  How are the different types of numbers related? For example, what about counting, whole, rational, irrational, integers, prime, ... ?

• Number Divisibility [Coltrain, 11/01/1998]
  How many natural numbers under 2,000 are divisible by 5 or 7? What's the quickest way to figure this out?

• Sets, Injection, and Surjections [McSweeney, 12/04/1998]
  If S is a set, such that every injection A from S to S is also a surjection, does S have the property that every surjection A from S to S is also an injection?

• Closure Property [Phillip, 12/22/1998]
  Simple definition and examples of closure property.

• Venn Diagrams [Remi, 01/24/1999]
  Members of a computer class choose at least one of three options. How many are taking just one? ... Use a Venn diagram.

• Closed Sets [Michael, 02/27/1999]
  Is a union of finite number of closed sets and the intersection of any number of closed sets closed?

• Subsets and Greatest Common Divisor [Megan, 03/26/1999]
  A question on subsets and another on greatest common divisor (GCD).

• Truth Table [Pretorius, 04/14/1999]
  Prove that the complement of X intersects Y = X' union Y' without using a Venn diagram.

• Dense and Nowhere Dense Sets [Patrick, 04/25/1999]
  Can you define the mathematical terms 'dense' and 'nowhere dense'?

• Unions and Intersections [Cole, 05/10/1999]
  How are the union and intersection of sets related?

• Stirling Numbers [John, 05/26/1999]
  Can you show how to evaluate Stirling Numbers of the first and second kinds?

• Mean of a Set of Numbers by Subsets [Bartlett, 08/15/1999]
  How can I prove that the mean of a finite set of numbers is the mean of the means of all the non-empty subsets of that set?

• Number of Unordered Partitions [Bonciocat, 08/18/1999]
  Is there a formula for the number of unordered partitions of a positive integer p(n)?

• Maximizing the Product of Partition Elements [Jacob, 08/20/1999]
  How can you prove that the product of the elements in a partition is at a maximum when the elements are equal?

• Total Membership [Foy, 08/20/1999]
  At a country club 35 people play golf, 28 swim, and 24 play tennis. Of these, 6 play golf and tennis only, 9 play golf and swim only, and 7 play tennis and swim only. 8 people do all three. How many members are there altogether?

• Infinity Hotel Paradox [Evening, 09/15/1999]
  How can a hotel with an infinite number of rooms, all already occupied, accommodate the passengers of an infinite number of buses without doubling them up?

• Is the Set of Complex Numbers Open or Closed? [Ricardo, 09/20/1999]
  Are the null set and C (the set of complex numbers) open sets, closed sets, both, or neither?

• Venn Diagram of Natural Numbers [Coons, 09/22/1999]
  How can I construct a Venn diagram comparing the numbers 1 through 100 in these 4 areas: odd, even, composite and prime?

• Unions and Intersections: Proving Sets [Edgar, 10/17/1999]
  How can I verify a proof of the statement A - (B union C) = (A - B) intersect (A - C)?

• Infinity as a Skolem Function [Zaba, 10/28/2000]
  Is infinity an absolute concept, a relative concept, or both?

• Injective, Surjective, Bijective Functions [David, 01/23/2001]
  What are the definitions of the terms injective, surjective, and bijective as they apply to functions in set theory?

• Linearly Independent Set Proof [Golban, 01/24/2001]
  Assume that in a vector space V, the vectors u and v are linearly independent. Prove that the set {2u-v, u+5v} is linearly independent.

• Countable Sets and Measure Zero [Jon, 05/12/2001]
  How would you prove that if a set S is countable, then S has measure zero?

• Set of Odds or Evens Bigger? [Danielle, 06/14/2001]
  Which is a bigger set of numbers, odds or evens?

• Sum of Integers in a Set [Martin, 06/19/2001]
  Consider the set of all four-digit integers, each of which is formed using the digits 1,2,3,5, or 7 at most once. Find the sum of the integers in this set.

• Closed Set of Elements [Craig, 06/30/2001]
  Please explain the term 'closed' in the following sentence: '...the set of complex numbers is CLOSED under addition...'.

• Sum of Set Numbers [Mark, 07/02/2001]
  a_1, a_2, a_3,....a_15 are positive real numbers... what is the smallest possible value of the largest of a_1, a_2, a_3...a_15?

• ONTO and INTO [Pawntep, 07/27/2001]
  What is the difference between ONTO and INTO when you describe a function?

• Subsets of Real Numbers and Infinity [Kevin, 08/22/2001]
  Am I correct in saying that both the whole number set and the integer set have an infinite number of numbers within them, and therefore are of the same size?

• Is an Empty Set a Subset? [Anabelle, 08/31/2001]
  For any event in the sample space, what is the proof that the empty set is a subset?

• Cardinality between Open and Closed Sets [Phillip, 09/20/2001]
  I would like to know how to prove that the sets (0,1) and [0,1] have the same cardinality.

• What is a Subset? [Phillip, 10/13/2001]
  I would like to know what a subset is.

• Unknown Numbers and a Venn Diagram [Christina, 11/26/2001]
  The GCF of two numbers is 20 and the LCM is 840. One of the numbers is 120. Explain how to find the other number and use the Venn diagram method to illustrate.

• How Many People Went on the Cruise? [Jessica, 12/03/2001]
  At the end of a special cruise, the employees could not remember the total number of people who were on board. However, they had the following data from the passenger list: 520 European females...

• Orders of Infinity [Steve, 12/05/2001]
  I recently read a book about infinity which set forth several arguments for why there are different sizes or orders of infinity. None of them seem convincing to me...

• Equivalence Relations [Joelle, 12/10/2001]
  Let X={1,2,3,4,5}, Y={3,4}. Define a relation R on the power set of X by A R B if A U Y = B U Y. Prove that R is an equivalence relation. What is the equivalence class of {1, 2}? How many equivalence classes are there?

• Line or Ray Longer? [Leslie, 12/11/2001]
  Which is longer, a ray or a line?