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Browse High School Sets
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Selected answers to common questions:
    Venn diagrams.

Definition of Equal and Equivalent Sets [05/31/2004]
Write a set that is equivalent to, but not equal to, the set (a, b, c, d, e, f). Would the answer be all the same letters, just arranged in a different order?

Definition of Set and How the Empty Set Fits within It [07/18/2004]
A set is defined as ďA collection of well-defined objects," and my students have a hard time understanding how a collection can be empty in the case of the null set.

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

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

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

Derfs and Enajs: Algebra and Venn Diagrams [03/09/2003]
All Derfs are Enajs. One-third of all Enajs are Derfs. Half of all Sivads are Enajs. One Sivad is a Derf. Eight Sivads are Enajs. The number of Enajs is 90. How many Enajs are neither Derfs nor Sivads?

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

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

Does Aleph-null Share Indeterminate Qualities with Infinity? [10/19/2004]
If I use a set N to represent the cardinality of an infinite set, then I imagine that N takes on the attributes of infinity. Thus, I wonder whether it is correct to associate the indeterminate qualities of infinity with aleph-null?

Duplicate Elements in Mathematical Sets [08/31/2004]
A discussion of what constitutes duplicate elements in a set and how set definitions apply to them.

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

Encrypting Many as One, and Decrypting Them Again [08/24/2014]
An adult seeks to encode a table of values into one number, with full recoverability. Taking a cue from random number generators, Doctor Douglas suggests a decimal representation, interleaving, and parsing protocol.

Equinumerous Sets [03/06/2003]
I'm trying to show that the set of real numbers and the set of irrational numbers are the same size.

Equivalence Relations [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?

Equivalence Relations [05/06/2003]
Determine with proof which of the three equivalence relation properties hold for the following relation...

Equivalence Relations [08/12/2003]
If I have a function that maps from R to R (where R is the reals) and a set of functions, say O(f), for g where there exist constants c and a in R+ (positive reals), where |g(x)| is less than or equal to c| f(x)| for x greater than a, how would I define a relation, say Q, on the set of functions mapping R to R by putting (g,h) in R iff g-h is in O(f)?

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

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

Five-Set Venn Diagram? [11/25/2001]
What does a five-set Venn diagram look like?

From a Point on a Line, to a Point in the Plane, to ... the Axiom of Choice [01/26/2011]
A student wonders about a bijective mapping of a point on the real number line to a point on the two-dimensional real plane -- and about the Axiom of Choice. With troubleshooting assistance from Doctor Jacques, Doctor Vogler offers maps and work- arounds, then provides a context for axioms.

From Reduction to Induction [02/03/2011]
Replace any two numbers x and y from (1, 2, ..., n) with the new single quantity x + y + xy; continue in this way until only one number remains. To find a formula for the smallest number possible from this procedure, Doctor Jacques lays the groundwork for a proof by induction.

Function Open but Not Continuous [03/06/2003]
Find a map R to R that is open but not continuous.

Functions and Inverses [05/29/2002]
Find sets A, B, and C, and functions f:A->B and g:B->C, such that (g o f) is both injective and surjective, but f is not surjective and g is not injective.

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

How Many People Went on the Cruise? [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...

How to Show 'Not' Statements on a Venn Diagram [05/03/2004]
How can I use a Venn diagram to display the opposite of a union or intersection of P & Q? For instance, how do I show (not P) or (not Q) versus not (P & Q)?

Inclusion-Exclusion Principle [09/03/2002]
In a survey of 100 people, 85 like calypso and 93 like pan. Calculate the number of people who like both calyso and pan.

Induction Proof [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.

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

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

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

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

Infinity Hotel Paradox [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?

Infinity, Zero [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?

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

Interval as Intersection of Sets [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.

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

Is the Number Line Both Continuous and Porous? [05/13/2005]
The number line is said to be densely populated because between any two numbers are infinite other numbers. But aren't there also an infinite number of gaps in the number line, with a tiny gap on either side of each of those discrete infinite numbers?

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

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

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