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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 {2u-v, 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 non-empty 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 one-to-one 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 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 set-builder 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, anti-symmetric, 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 (A-B)-C = (A-C)-(B-C), 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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