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 TOPICS This page:   sets    Search   Dr. Math See also the Internet Library:   sequences/sets and   set theory HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Sets Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Venn diagrams. Laws of Arithmetic [02/12/1999] The Distributive Law, Associative Law, Identity, and Commutative Equations. 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 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 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? Page: []

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