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 TOPICS This page:   logic/set theory    Search   Dr. Math See also the Dr. Math FAQ:   false proofs,   classic fallacies Internet Library:   logic and   set theory COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Logic and Set Theory Stars indicate particularly interesting answers or good places to begin browsing. Infinite Sets [09/24/1997] In my algebra class we have been debating whether the integers or the whole numbers contain more elements... Integer Logic Puzzle [04/22/2001] Two integers, m and n, each between 2 and 100 inclusive, have been chosen. The product is given to mathematician X and the sum to mathematician Y... find the integers. Inverses within Semigroups [05/06/2002] I would like to know the value of e in a semigroup for exponentiation, if it exists. In other words what CONSTANT value e satisfies the equation, a^b = e? Is There a Universal Set of All Numbers? [06/16/2004] The real numbers and the imaginary numbers are subsets of the complex numbers. Is the set of complex numbers a subset of a more universal set? Is there a universal set of all numbers agreed upon today? Karnaugh Maps [05/07/2000] What are Karnaugh maps? How are they used? Lewis Carroll's Logic Problems [01/15/1997] Where can I find out more about Lewis Carroll's logic problems? 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. Logical Sentences and Logical Arguments [05/16/2008] What is the difference between A |- B and A -> B? They seem to mean the same thing--if A is true then you know that B is also true. Logic and Conditional Sentences [10/04/2005] I am having a hard time understanding why two false statements in a conditional sentence makes it true. Logic: Bayes and Popper [06/24/2003] Is p -> q totally equivalent to ~q -> ~p in practice? Mathematical Logic [02/09/2001] Assumptions, rules, contradictions, and a derivation. Math Symbols [04/07/1997] What do the common math symbols (backward E, upside-down A, etc.) mean? The Meaning of 'Or' in Logic Statements [12/19/2003] If a logic statement says, 'James is taking fencing or algebra,' does that mean he is taking one class or the other, or could he be taking both of them? Necessary and/or Sufficient [05/26/2002] What does it mean to say that a condition is necessary, sufficient, or necessary and sufficient? Necessary and/or Sufficient Conditions with Modular Math [12/01/2006] I'm working on a question in modular math that asks me to identify whether given conditions are "necessary", "sufficient", or "necessary and sufficient". I'm not sure what those terms mean. One-to-One Correspondence of Infinite Sets [03/26/2001] How can I prove that any two infinite subsets of the natural numbers can be put in a 1-1 correspondence? Order of Quantifiers [12/19/2002] Can you help me understand the order of quantifiers? 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... Ordinals: Sets or Numbers? [04/16/2010] Are ordinals sets or numbers? Doctor Tom resolves confusion around how to think about ordinals by putting them in the context of Zermelo-Fraenkel (ZF) set theory. Problem from Real Analysis [10/05/2002] Let X = A U B where A and B are subspaces of X. Let f:X->Y. Suppose that the restricted functions f|A:A->Y and f|B:B->Y are continuous. Show that if A and B are closed in X, then f is continuous. Proof by Contradiction [04/29/2003] Is there any specific mathematical theory that states that Proof by Contradiction is a valid proof? Proof of One Step Subgroup Test [05/13/2002] Prove that a nonempty subset H of a group G is a subgroup of G if and only if a*b^(-1) is in H for all a, b in H. Proof Styles--Contradiction and Direct [04/22/2006] Under what circumstances is it easier to prove a mathematical problem by the method of contradiction versus direct proof? I know the contradiction proof that sqrt(2) is irrational. Can you prove that by direct deduction? Proof that 1 + 1 = 2 Using Peano's Postulates [09/12/2002] How do I prove that 1 + 1 = 2? Proof that f(K) is a Subgroup of G' [11/26/2001] If G and G' are groups, f is an isomorphism from G into G', and K is a subgroup of G, then show that the set f(K)={f(k)\k is a member of K} is a subgroup of G'. Properties of Relation [05/28/2003] What are reflexive, symmetric, anti-symmetric, and transitive relations? star, please Proving a Topology [10/11/1998] Let X be an uncountable set of points, and T consist of the empty set and all subsets of X whose complement is finite. Prove that T is a topology of X. Russell Paradox [7/2/1996] I'm looking for a demonstration of the Russell Paradox (there is no ensemble of all ensembles). Sets N, R, C, Z, and Q [01/22/2001] What are the exact and extensive definitions of the sets N, R, C, Z and Q? What relation do these sets bear to one another? Significance of Rational Numbers [01/11/2003] Why are rational numbers defined the way they are? Solving the Equation x^y = y^x [12/09/2004] Solve x^y = y^x for x in terms of y only. Also, how do I find all possible solutions beyond the obvious ones of x = y, (2,4), and (4,2)? Subsets and Subspaces [09/07/2001] I'm trying to extract some relations in set notation from a description from Loomis and Sternberg's _Advanced_Calculus_... Sum of Uncountably Many Positive Numbers [09/15/2004] Let S be a set of uncountably many positive numbers. I would like to show that the sum of all the elements in S is infinite. That is, there is no convergent sum of uncountably many positive terms. Transfinite Numbers [11/07/1997] I know that Georg Cantor discovered transfinite numbers, but what are they? Truth of a Biconditional Statement [11/08/2005] Let p represent x = 0, and let q represent x + x = x. Write the biconditional p <-> q in words. Decide whether the biconditional is true. Truth of the Contrapositive [06/07/2003] The inverse of a statement's converse is the statement's contrapositive. Why? Understanding the Transitive, Reflexive, and Symmetric Properties [06/30/2008] Decide if the relation 'is not equal to' is a)transitive, b)reflexive, and c)symmetric with regard to the counting numbers. Page: [

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