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 TOPICS This page:   logic    Search   Dr. Math See also the Dr. Math FAQ:   false proofs,   classic fallacies and   liars, truthtellers and   proofs Internet Library:   logic 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 Logic Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Venn diagrams. Parts of a Biconditional Statement [06/03/1999] Does the "necessity" condition correspond to "only if" and "sufficient" correspond to "if," or is it the other way around? Party Guests and Perfect Squares [07/02/2001] Who was dancing with whom? Philosophy of the Truths of Mathematics [02/28/2001] Do the truths of math hold in any conceivable world? The Prisoners' Dilemma [12/8/1995] I'm looking for a paper - or some material - about "the prisoners' problem." Probability of Two Male Children [7/5/1996] If a family has two children, and the older child is a boy, there is a 50 percent chance the family will have two boys. However... Projects on Puzzles or Mazes [11/13/2002] I would like to do a project that involves applying mathematics to areas like puzzles or mazes. Proof by Contradiction [04/29/2003] Is there any specific mathematical theory that states that Proof by Contradiction is a valid proof? Proof by Contradiction: A Monkey's Uncle [12/09/1998] How can I verify a mathematical fact by using a proof by contradiction? Proof Involving Even and Odd Numbers [02/08/2005] Prove that if a - 3b is even, then a + b is even. Proofs Using Quantifiers [10/17/2001] I have no idea how to do this proof... Proof that 1 + 1 = 2 Using Peano's Postulates [09/12/2002] How do I prove that 1 + 1 = 2? Properties of Relation [05/28/2003] What are reflexive, symmetric, anti-symmetric, and transitive relations? star, please Propositional Logic and English [03/04/2003] Given the propositions: P: The sun shines; Q: The wind blows; R: The rain falls; S: The temperature rises; write in words the compound proposition Q Xor R (Q Exclusive-Or R). Propositional Logic: Translating "You cannot ride the roller coaster if ... unless ..." [08/06/2009] Translate this into propositional logic: "You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old." Proving Conditional Probabilities [01/24/2001] If P(B|A) is greater than P(B), prove that P(B^c|A) is less than P(B^c). Proving Mathematical Induction is Correct [08/31/2001] I know that I can use proof by contradiction, but I do not know how to start. Quantifiers [03/25/2003] Four problems about quantifiers. Rational and Irrational Numbers: Multiplication, Division [10/15/2001] I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational. Red and Blue Hats [03/20/2001] Alan, Ben, and Cal are seated, with their eyes closed. Three hats are placed on their heads from a box that contains three red and two blue hats... Red Hat, Blue Hat [11/16/2002] A teacher puts one hat on each of three students' heads and then discards the remaining two hats so they cannot be seen. Then the first child is told he can look at the other two children and from the color of their hats, he can guess what color he is wearing... 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? Sequence of Integers [08/12/2008] Find all functions f such that for each n in Z+ we have f(n) > 1 and f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18. Sets and Elements [03/29/2003] Find the elements of the Set A = {{1,2,3},{4,5},{6,7,8}} and determine whether each of following is true or false... 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? Sets, Subsets, and the Null Set [06/22/2004] I know that the null set is a subset of any set, but can the null set be an element of a set? Sick Monks Leave the Monastery [09/21/2002] Sick monks must leave the monastery. Each has a red cross on his forehead. After three days, all the sick monks have left. How many sick monks were there in the monastery? Simple Proof by Induction [08/27/1999] How can I show by mathematical induction that the proposition "if n >= 1 then 3n >= 1 + 2n" is true? sin(sqrt2) More Simply? [05/17/2017] A teen seeks simpler forms for trigonometric functions of radicals. With examples, Doctor Vogler explains how this pursuit leads to infinite cardinalities of sets and further number theory. Solving Axiom Systems [02/11/2003] I have to prove the theorems with the axioms and I have no idea how to do it. Solving Questions [08/28/2002] In a poll of 34 students, 16 felt confident solving quantitative comparison questions, 20 felt confident solving multiple choice questions.... How many students felt confident solving only multiple choice questions and no others? 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)? Some Drank All; Did All Drink Some? [02/23/2017] Three choices lead to too many overlapping preferences. Drawing a Venn diagram, Doctor Peterson fills in some regions with given information, then begins subtracting to deduce intersections and unions. A Statement to Save a Life [05/28/1997] If you lie, you will die by poison; if you tell the truth, you will be shot. What statements will save your life? Sum of An Infinite Series [07/08/1998] Is it possible to add up all the terms of an infinite series? Tautologies and Contradictions [02/19/2006] Is there a way to tell if a logic statement is a tautology or a contradiction without having to make a truth table? Tautologies in Logic Proofs [02/14/2001] Can you do a proof in which the conclusion or one of the hypotheses is a tautology? Aren't all proofs tautologies? Tautology and Substitution Principles [02/28/2001] What is the difference between the tautology principle and the substitution principle? Tiling with Dominoes [08/06/2001] A 6-square by 6-square board is tiled completely with 18 2x1 dominoes. Prove that at least one horizontal or vertical line can be drawn along the edges of the dominoes that divides the board into 2 regions, without cutting any dominoes in half. Transfinite Numbers [11/07/1997] I know that Georg Cantor discovered transfinite numbers, but what are they? Page: []

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