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Browse High School Sets
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Venn diagrams.
 Linearly Independent Set Proof [01/24/2001]

Assume that in a vector space V, the vectors u and v are linearly
independent. Prove that the set {2uv, u+5v} is linearly independent.
 Linear Topology [02/09/2003]

If a point in set X is finite, then X has a first point and a last
point. Prove by induction if true, and give a counterexample if false.
 Line or Ray Longer? [12/11/2001]

Which is longer, a ray or a line?
 Maximizing the Product of Partition Elements [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?
 Mean of a Set of Numbers by Subsets [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 nonempty subsets of that set?
 Measurable Functions [09/13/2004]

Can you provide some motivation for the definition of a measurable
function?
 Noting Nothing: Braces around Zero vs. Empty Braces [08/02/2010]

A student struggles to distinguish among the meanings of {0}, {}, and the symbol phi.
Doctor Ian clears up the set theory notation, then emphasizes that the empty set can,
itself, be an element.
 Null Sets [8/23/1996]

Is the null set a proper subset of every set?
 Number Divisibility [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?
 Number of Unordered Partitions [08/18/1999]

Is there a formula for the number of unordered partitions of a positive
integer p(n)?
 Number Systems: Two Points of View [06/30/1998]

What are the number systems?
 One to One Correspondence Between Sets [04/04/1997]

What is the onetoone correspondence between the set of natural numbers
and the set "S" = {1/3, 1/6, 1/12, 1/24, 1/48, ...}?
 ONTO and INTO [07/27/2001]

What is the difference between ONTO and INTO when you describe a
function?
 Open Balls and Open Sets [10/04/1998]

Can you explain the concepts of open balls and open sets? What about in
the plane (2D) and in space (3D)?
 Open Sets and Metric Spaces [11/17/2002]

(M,d) is a metric space. Prove Ø is open; prove M is open.
 Orders of Infinity [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...
 Origin of the Null Symbol [03/05/2003]

Where did the null symbol come from?
 Overlapping Sets [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?
 Partitioning Sets into All Possible Subsets [05/19/2003]

I am trying to determine how to partition a set into all possible
subsets, using all items in the set in each combination of subsets.
 Percentage of Numbers Divisible by 6, 8 [03/18/2003]

What percentage of numbers is divisible by 6? by 8?
 Polynomial Degrees and Definition of a Field [03/02/1998]

The degree of polynomials added together, and definition of a field.
 Positive Numbers Less Than 3 [01/30/1998]

Write the set C of positive multiples of 3 less than 3 using roster
notation and setbuilder notation. How can a positive number be less than
3?
 Probability of Random Samples by Complementarity, Inclusion, and Exclusion [08/09/2012]

Given a set S = {A_1, A_2, A_3, ..., A_N}, a probability student struggles to enumerate
all cases of random samples of size k that contain at least its first two elements, A_1
and A_2. Doctor Schwa provides hints based on complementary probability and the
principle of inclusion and exclusion.
 Proof Involving Rational and Irrational Numbers [09/13/2004]

Prove the existence of 2 irrational numbers, a and b, where a^b is
rational.
 Proof: Rational q and Irrational p [03/03/2003]

Prove that if a and b are rationals, then there is a rational q and an
irrational p so that both p and q are between a and b.
 Properties of Relation [05/28/2003]

What are reflexive, symmetric, antisymmetric, and transitive
relations?
star, please
 Prove S' Closed in R^n [10/07/2002]

Prove that S' (the derived set or set of all accumulation points of S)
is closed in R^n.
 Prove That a Set Is Uncountably Infinite [10/31/1997]

Prove that the set [0,1]x[0,1] is uncountably infinite.
 Proving an Equivalence Relation [01/31/2003]

Let R1 and R2 be equivalence relations on sets S1 and S2,
respectively. Define a relation R on S1 X S2 (Cartesian Product) by
letting (x1, x2)R(y1, y2) mean that x1R1y1 and x2R2y2. PROVE that R is
an equivalence relation on S1 X S2, and describe the equivalence
classes of R.
 Proving a Set is Closed [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.
 Raising Sqrt(2) [10/14/1997]

What is the value of x in the following equation: x^x^x^x^x ... = 2 ?
 The Real Number System in a Venn Diagram [09/08/1998]

How do you construct a Venn diagram that illustrates the real number
system?
 Relations on a Set, as Mappings [7/19/1996]

Proof: If R, S, and T are relations on a set A, show that (R o S) o
T = R o (S o T), where "o" stands for composite...
 Russell's Infinite Set Paradox [03/25/1998]

Given the set (S) of all sets that do not contain themselves, does S
contain itself?
 Set and Element Relations [03/23/2003]

On a set of n elements, how many relations are there that are
reflexive and antisymmetric? irreflexive and symmetric?
 Set Equality [10/12/1998]

Can you help me show that (AB)C = (AC)(BC), where A, B, and C are
sets?
 A Set Function F and Its Inverse F(1) [03/24/2003]

Injective and surjective set functions.
 Set of Odds or Evens Bigger? [06/14/2001]

Which is a bigger set of numbers, odds or evens?
 Sets: Cardinality and Functions [06/29/1998]

What does "{1,0}^{a,b}" mean?
 Sets Containing an Infinite Number of Members [02/03/1998]

Which of these two sets is larger  the set of all rational numbers, or
the set of all irrational numbers?
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