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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.



Cantor, Peano, Natural Numbers, and Infinity [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 [01/08/1997]
How can 3 be both a cardinal and ordinal number at the same time?

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

Diagram for Math Numbers [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?

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

Infinite Sets [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, Difference, Union [12/18/2002]
Please explain: integers that are members of A but not of B, integers that are members of both A and B, and integers that are members of either A or B?

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

Line Segments and Size of Infinites [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 [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 [12/10/1999]
Can you explain what Cantor meant by one-to-one correspondence, and transfinite numbers?

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

Rational Numbers [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?

Sets and Subsets [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?

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

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

Set Theory and Orders of infinity [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?

Unions and Intersections [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? [09/06/2001]
Can you please give me examples of sets and subsets?

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

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

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

Are They Wearing Seatbelts? [3/26/1995]
80 percent of all California drivers wear seatbelts. If 4 drivers are pulled over, what is the probability that all 4 will be wearing their seatbelts?

Bijections [04/28/2003]
Find a bijection from (0, 1) to (0, 1]. (Be sure to prove that your function has the proper properties.)

Borel fields [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.

Bounded Set [06/24/2003]
Let S be a set of real numbers. Prove that the following are equivalent: (a) S is bounded, i.e. there exists a number M greater than 0 such that abs(x) is less than or equal to M for every x in S; (b) S has an upper bound and lower bound.

Building Sets [05/26/2002]
Is 5 part of the set {x:x is a multiple of 7 and 5 < x < 56}?

Cantor's Infinities and Universal Sets [07/17/2008]
In Cantor's set theory, the idea of having a universal set or a set of everything cannot be true, due to the basic contradiction that arises from the nature of set theory. Based on this, when looking at Cantor's Hierarchy of Infinities, does the hierarchy of infinities still hold? Truly, if an absolute infinity existed then it would accommodate everything, contradicting the idea of no universal set.

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

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

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

Closure and Compactness in a Metric Space [10/08/2002]
Regard Q, the set of all rational numbers, as a metric space, with d(p,q)=|p-q|... Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q?

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

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

Compact Sets and Hausdorff Spaces [03/19/2003]
How do you prove that every compact subset of a metric space is closed?

Complement of a Set [08/14/2003]
What is a universal set?

Complements, Unions, and Intersections of Sets [10/13/2004]
A visual example of how to shade in the complement, union, and intersection of sets in a Venn diagram.

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

Countability of Primes and Composites [05/18/2002]
If the union of two sets is countable, can either of the sets be uncountable?

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

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