Logic

• Cantor, Peano, Natural Numbers, and Infinity [Bennett, 03/19/1998]
  A conversation on transfinite numbers and contradictions the questioner believes exist in Cantor's paper introducing the diagonal method.

• Contrapositive, Converse, Inverse [Goode, 06/10/1999]
  How can I write the contrapositive, converse, and inverse of and prove or disprove the statement, "If m + n is even, then m and n are even"?

• A False statement Implies Any statement [Kemp, 09/25/1997]
  Do you know any ways to make learning and remembering quantifiers and laws of inference easier?

• Finding One Coin of 12 in 3 Steps [Schwartz, 8/6/1996]
  Given a pile of twelve coins of equal size with one of a different weight, in three weighings find the unequal coin...

• Inductive and Deductive Examples [Mehrdad, 11/21/2001]
  Solve by deduction and then induction: Bob wants to figure out what his teacher wants for his birthday, but he cannot ask his teacher directly. How does he pick the perfect present?

• Logic: Definitions [Jenn, 04/04/2000]
  What does deductive reasoning mean? What does inductive reasoning mean?

• Logic of Indirect Proofs [Carnrike, 10/16/1996]
  Can you explain the logic of indirect proofs?

• Lucky Seven Fractions Puzzle [Victor, 12/22/2001]
  Put numbers 1-9 in order to make the equation correct: XX/XXX+XX/XX=7.

• Main Connectives in a Proof [Elizabeth, 10/28/2001]
  Focusing on the main logical symbols in a proof.

• Probability: Let's Make a Deal [Gordon, 4/29/1996]
  Should the contestant stick with the original choice of doors or switch and choose the other door? What about the lottery?

• Properties and Postulates [Nwasokwa, 08/04/1999]
  How do you discover or create a property? What is the difference between a property and a postulate? Do we have to prove all properties?

• The Sportsville Teams [Isak, 10/30/1997]
  Are there straightforward methods for solving logic problems?

• Truth Tables and Computer Circuits [VanEtten, 01/17/2000]
  Can you please explain how to read and draw computer circuit diagrams, how to form truth tables from reading the diagrams, and the logical arguments behind this?

• The Truth-teller, the Liar, and Ambiguous [Clerx, 7/9/1996]
  God knows everything and always replies the truth. The devil knows everything, but always lies. The third person's answers are completely useless and could be right or wrong. Using three questions, determine who is who.

• What is Fuzzy Logic? [Chaney, 10/29/2000]
  I need a simple definition of fuzzy logic and what it's used for.

• Where and with What Weapon Was Each Man Murdered? [Sarah, 09/19/2001]
  One rainy evening, five military men were murdered in the old mansion on Willow Lane... use the clues to make a grid and find who was murdered with what weapon, and where.

• Who Owns the Zebra? [Joseph, 08/05/1997]
  I have tackled this indirect reasoning problem and have come up with a solution that I am not sure is correct.

• Who Stole Second Base? [Willie, 2/20/1995]
  The umpire was convinced that either Archie, Buster, Cal, or Dusty had stolen second base. Each player, in turn, made a statement, but only one of the four statements was true. Who told the truth?

• Monty Hall Strikes Again [Pryor, 11/2/1994]
  There are three cups, one of which is covering a coin. I know the whereabouts of the coin, but you don't. You pick a cup, and I take one of the remaining cups, one which DOESN'T contain a coin. Both you and I know the cup I pick doesn't contain a coin. You then have the option to swap your cup with the third, remaining cup, or keep your first choice. What is the probability of the coin being in the cup if you keep your first choice, or if you decide to swap them?

• A Truel [Zechiel, 3/6/1995]
  You and two of your friends get into a dispute and decide to solve it with a "Truel", a three way duel.

• A Geometry and a Logic Problem [Marsville, 3/11/1995]
  Problem 1: A cylindrical hole six inches long is drilled straight through the center of a solid sphere. What is the volume remaining in the sphere? Problem 2: The classical stay-switch problem.

• Logic - Liars & Truthtellers (What Question Does She Ask?) [Marsville, 3/12/1995]
  A logician vacationing in the South Seas finds herself on an island inhabited by the two proverbial tribes of liars and truth-tellers.

• Zeno's Paradox [Gut, 10/19/1995]
  At eleven o'clock I put ten balls numbered 1,2, ...10 in a box and immediately take out the ball numbered 1. At eleven thirty I put balls numbered 11 through 20 into the box and take out the ball numbered 2. At eleven forty-five I put balls numbered 21 through 30 into the box and take out the ball numbered 3. This continues at time intervals that are half of the preceding one. How many balls are in the box at twelve o'clock?

• The Prisoners' Dilemma [Tim, 12/8/1995]
  I'm looking for a paper - or some material - about "the prisoners' problem."

• Geometry Puzzles [Creswell, 12/18/1995]
  A student asks Dr. Math for help in finding the correct combinations of numbers to solve two puzzles.

• Which Twin is Telling the Truth? [Sarah, 3/2/1996]
  At a fork in a road are identical twins. One always lies and one always tells the truth, but you don't know which is which. If you could only ask one question to find out which way to go, what would it be?

• Where is the Prize? [Aircool, 4/17/1996]
  "One of these three boxes is filled with precious jewels. The other two contain pebbles from a pond..."

• What Color Hat am I Wearing? [Yu, 4/17/1996]
  Three students close their eyes, and the teacher puts a hat on each of their heads (hiding the other two hats)...

• Math Logic [Marone, 6/5/1996]
  Sally, Ron, Jim, and Meghan are President, VP, Treasurer, and Captain of the cheerleading squad, but not necessarily in that order. Who is what?

• Was Henry Guilty? (Geometry Puzzle) [Stevens, 6/10/1996]
  In Hughmoar County, residents shall be allowed to build a straight road between two homes as long as the new road is not perpendicular to any existing county road...

• Probability of Two Male Children [McClory, 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...

• Negation in Logic [Thomas, 8/3/1996]
  What is the negation of "In every village, there is a person who knows everybody else in that village"?

• Football Logic: Who Defeated Whom...? [topp, 8/17/1996]
  Four teams were brown, blue, red, and purple; the competing captains were Albie, Barry, Bill, and Ben... who defeated whom?

• Who is a Liar, Who Tells the Truth? [Minard, 9/4/1996]
  Swimmers always tell the truth, non-swimmers always lie. If you meet three such people, and ask them... which of these people is a swimmer or non-swimmer?

• Invalid Logic Argument [Archer, 9/9/1996]
  If I want the result to be true but the premise is false, the argument will be invalid...

• Lewis Carroll's Logic Problems [Constantinescu, 1/15/1997]
  Where can I find out more about Lewis Carroll's logic problems?

• Logical Equivalents [Columber, 3/3/1997]
  Prove or disprove: existential x P(x) and existential x Q(x) is logically equivalent to existential x (P(x) and Q(x)).

• Fuzzy Logic [Graham, 04/26/1997]
  What is fuzzy logic?

• A Statement to Save a Life [Ray, 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?

• Goedel's Incompleteness Theorem [Seldon, 08/08/1997]
  How did Goedel prove that any nontrivial logical system cannot be proven to be inconsistent?

• If P then Q [Jarchafjian, 08/29/1997]
  I don't understand how if p is false then regardless of q the statement is true.

• Indirect Proofs [Ashley, 09/21/1997]
  If Clark is a mathemagician, then Lois is his assistant...

• How Many Pieces of Candy in Each Jar? [Hemstock, 09/30/1997]
  At the annual Cumberland County fair, one of the more popular booths is the Candy Contest...

• Two Turkeys [Guilmain, 10/07/1997]
  ... How much did each turkey weigh?

• Boolean Algebra Problems [Fleming, 12/05/1997]
  Prove x'y' + x'y + xy' = x' + y'; x'y' + x'y + xy' + xy = Identity.

• The Four Doors of Xanth [Goodwin, 02/11/1998]
  Each door conceals one item: a treasure, a rope, a key, and a lantern. You must find all four items in a particular order to keep the treasure.

• Coat Confusion [Maxwell, 03/20/1998]
  When the fire alarm went off, 6 people in a room each grabbed a coat, but no one took his own. Who took A's coat?

• Russell's Infinite Set Paradox [McAllister, 03/25/1998]
  Given the set (S) of all sets that do not contain themselves, does S contain itself?

• Paradox of the Unexpected Exam [McDaniel, 03/26/1998]
  A teacher announces that a test will be given next week on one of the five weekdays. Why won't the test ever be given?

• Two Mathematicians Problem [Tuncel, 05/18/1998]
  One mathematician is give the sum of integers X and Y, and another is given their product... what are the numbers?

• Computers: Defining Logical Operations [Towella, 05/21/1998]
  Can you tell me the meaning of the following logical operations: AND, OR, XOR, NAND, NOR, NOT?

• Mathematical Induction [Teo, 07/01/1998]
  Proof by induction does not prove anything, because in the inductive step, one makes the assumption that P(k) is true...

• Sum of An Infinite Series [May, 07/08/1998]
  Is it possible to add up all the terms of an infinite series?

• Knowing People at a Party [Angeles, 08/27/1998]
  Prove that at any party, there are two people who know the same number of people. Assume that if A knows B, then B knows A. Assume also that everyone knows himself or herself...

• Explaining Mathematical Induction [Caruselle, 08/29/1998]
  What is mathematical induction? I have to do a report on it.

• John Venn and Venn Diagrams [Potemra, 09/04/1998]
  Can you give me some information on John Venn and the origin of Venn diagrams?

• Mathematical Induction [Katie, 09/07/1998]
  What is mathematical induction? Can you give an example of the ideas of math induction?

• Who Picked the Most? [Will, 09/12/1998]
  Arrange the names of the people in the order of the number of peaches that each picked, starting with the person who picked the most.

• Disjunctive Syllogism [Ray, 09/25/1998]
  What is the Disjunctive Syllogism?

• The Truel [Sabina, 10/13/1998]
  A truel is a duel with three participants, rather than two. Whom should Mr. Black shoot first to survive?

• Negating Statements [Heather, 10/27/1998]
  What is negation? What is a statement? How do you negate a statement?

• Game Theory and Payoff Matrices [Haldar, 11/14/1998]
  Can you give me a good introduction to game theory? What is a payoff matrix?

• Proof by Contradiction: A Monkey's Uncle [Schmidt, 12/09/1998]
  How can I verify a mathematical fact by using a proof by contradiction?

• Gauss' Method for Solving Equations [Jacob, 12/11/1998]
  How do you use Gauss' method and matrices to solve systems of equations? Why does this method work?

• Who Made Which Toys? [Katie, 12/21/1998]
  A math logic problem, from a rhyme describing Santa's toymakers.

• Godel's Incompleteness Theorem [Andersen, 01/18/1999]
  What does Goedel prove in his incompleteness theorem?

• Logic, Groups, and Identities [Anna, 02/25/1999]
  Is it possible for more than one answer to exist when proving things? What is a group? Can you give an example of an identity?

• Math Logic - Determining Truth [De Hamer, 04/13/1999]
  A number divisible by 2 is divisible by 4. Find a hypothesis, a conclusion, and a converse statement, and determine whether the converse statement is true.

• Parts of a Biconditional Statement [Abdellah, 06/03/1999]
  Does the "necessity" condition correspond to "only if" and "sufficient" correspond to "if," or is it the other way around?

• Finding the Operation [Yehling, 06/21/1999]
  How can I find the operation '?' given 3?4 = 5, 4?7 = 1, 8?4 = 0, and 1?2 = 9?

• Simple Proof by Induction [Bourouba, 08/27/1999]
  How can I show by mathematical induction that the proposition "if n >= 1 then 3n >= 1 + 2n" is true?

• Boolean Algebra Proofs [Perego, 09/25/1999]
  Prove the Boolean expression ab + bc + ca' = ab + ca'; also, prove using contraposition that 2(q^2) does not equal (p^2) when p and q are relatively prime.

• Unions and Intersections: Proving Sets [Edgar, 10/17/1999]
  How can I verify a proof of the statement A - (B union C) = (A - B) intersect (A - C)?

• Largest x, x^2 less than 2 [Michael, 10/23/1999]
  Prove that there is no largest real number x, such that x^2 is less than 2. (Use indirect proof.)

• Karnaugh Maps [Goold, 05/07/2000]
  What are Karnaugh maps? How are they used?

• Basic Truth Tables and Equivalents in Logic [Carissa, 05/23/2000]
  What are the truth tables for basic propositional logic operations? What are some useful equivalencies?

• De Morgan's Laws [Placke, 09/21/2000]
  What is "De Morgan's Law"?

• Deductive Reasoning [Toi, 09/21/2000]
  What is deductive reasoning? How do you use it?

• Conjunctive and Disjunctive Normal Forms [Patrick, 09/25/2000]
  How can I find the conjunctive normal form (CNF) of an expression from the disjunctive normal form (DNF)?

• Introduction to Logic and Truth Tables [Carolyn, 09/27/2000]
  I can't figure out the p and q thing. Can you explain what they are and how operations like "AND" work?

• Constructing Truth Tables [Zertuche, 11/03/2000]
  How can you make a truth table for the expression pv(p^~q)?

• The Indeterminate Nature of 0/0 [Rob, 12/21/2000]
  I have a theory that 0/0 = any number, and is not "indeterminate" as is traditionally claimed. Can you explain the flaw in my thinking, and the "indeterminate" nature of 0/0?

• Choosing a Random Rational Number [Platt, 01/23/2001]
  0 percent probability should mean that it is impossible for a rational number to be chosen from the set of real numbers, but obviously this isn't the case. How can it be 0 percent probability but not impossible?

• XOR Operation [James, 01/23/2001]
  An explanation and a few examples of the XOR operation.

• Proving Conditional Probabilities [Walker, 01/24/2001]
  If P(B|A) is greater than P(B), prove that P(B^c|A) is less than P(B^c).

• Paradox and Fallacy [Lee, 01/25/2001]
  What is the difference between paradox and fallacy in mathematics?

• Logical Fallacies [Scavenger, 01/29/2001]
  Don't fallacious arguments (as in the argumentum ad ignorantiam) represent illogic?

• Fibonacci Proof [Klein, 01/29/2001]
  This proof is giving me major problems: F(2n) = (F(n))^2 + (F(n-1))^2. ...

• Mathematical Logic [Chin, 02/09/2001]
  Assumptions, rules, contradictions, and a derivation.

• Context, Language, and False Equations [Rheanna, 02/12/2001]
  Is there such a thing as a false equation? How does the context in which it is set affect the truth of an equation?

• Tautologies in Logic Proofs [Bryant, 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 [Sara, 02/28/2001]
  What is the difference between the tautology principle and the substitution principle?

• Philosophy of the Truths of Mathematics [Lauren, 02/28/2001]
  Do the truths of math hold in any conceivable world?

• Monty Hall Logic [Dinwiddie, 03/09/2001]
  Are there in fact four options? Aren't there three choice points, not just two?

• Truth Tables of Boolean Variables [Green, 03/09/2001]
  Given two Boolean variables, A and B, what are the sixteen possible truth functions in table form?

• Red and Blue Hats [Belvona, 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...

• Match Couples and Parties [Jesse, 03/28/2001]
  Read the clues given, and match everything up.

• Absorption Laws [Tony, 04/10/2001]
  Prove x + x'y = x + y, or x + xy = x.

• Paradox [Ariel, 05/07/2001]
  What is a paradox?

• Boolean Algebra and DeMorgan's Theorems [Craig, 05/14/2001]
  How do I simplify not{not[A and not(B)] or C}?

• Crossing the Desert [Lefebvre, 05/22/2001]
  A truck gets one mile per gallon, and can hold 400 gallons at a time. How much is the minimum amount to cross a 1000-mile desert?

• Integer Logic Puzzle [Andrew, 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.

• Truth Tables: And, Or, Implies, Not [Lisa, 06/10/2001]
  Can you give me an explanation of truth tables, with examples?

• Two- vs. Many-Valued Logic [Mikael, 06/19/2001]
  Are the perhaps practical instruments of "fuzzy logic" and not at least "three- or many-valued logic" for pragmatic use only?

• Exclusive or Inclusive Disjunction? [Gemma, 06/28/2001]
  I have difficulty interpreting this problem, especially the first sentence: Tanya is either a singer or a ballerina...

• Induction Proof with Inequalities [Jay, 07/03/2001]
  Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative integer.

• Party Guests and Perfect Squares [Kristy, 07/02/2001]
  Who was dancing with whom?

• Modus Ponens [Erin, 07/10/2001]
  A man born in 1806 is x years old at the year x squared. Solve for x.

• Inductive vs. Deductive Reasoning [Angie, 07/24/2001]
  Can you explain the difference between inductive and deductive reasoning?

• Proving Mathematical Induction is Correct [Soh, 08/31/2001]
  I know that I can use proof by contradiction, but I do not know how to start.

• Converse, Inverse, Contrapositive [Hana, 09/08/2001]
  Write the converse, inverse, and contrapositive of each conditional and determine whether they are true or false; if false, give a counterexample.

• Truth Tables: a AND b [Emilee, 09/10/2001]
  What are truth tables?

• Open Sentence, Statement [Kristin, 09/18/2001]
  What is an open sentence?

• Upside-down A's and Sideways E's [Chang, 09/19/2001]
  What is an upside-down A? a sideways E? Also, what do R and J stand for?

• Rational and Irrational Numbers: Multiplication, Division [Jess, 10/15/2001]
  I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational.

• Proofs Using Quantifiers [Melanie, 10/17/2001]
  I have no idea how to do this proof...

• Domain, Parentheses, and Brackets [Andrew, 11/01/2001]
  Translate: All people who are not poor and are smart are happy. Those people who read are not stupid. Jill can read and is wealthy. John is poor and stupid. Happy people have exciting lives....

• Indirect Proof of Parallel Lines [Ryan, 11/26/2001]
  I have asked my high school geometry class to prove indirectly that parallel lines have the same slope. Unfortunately, I cannot figure out how to do it myself...

• Binet's Formula and Induction [JT, 11/28/2001]
  What is induction, and can you prove Binet's formula by induction?

• Why p and q? [Laurie, 11/29/2001]
  We are wondering why the letters p and q are used to abbreviate statements.

• What is a Property? [Jim, 11/29/2001]
  I understand Undefined and Defined terms and Axioms and Theorems, but what exactly is a Property? Is it the same thing as a Theorem? Also what is a Law?

• Who Got Engaged to Whom? [Alina, 11/27/2001]
  Dorothy, Jean, Virginia, Bill, Jim, and Tom became engaged to one another. Who got engaged to whom?

• Who Gets the Job? [Pete, 12/11/2001]
  Each job candidate can see the other two candidates' black or red dots but not his own. Whoever can figure out the color of his own dot gets the job.

• All People in Canada are the Same Age [Kathleen, 01/18/2002]
  For n = 1,2,3,..., every bag contains n solid-colored balls of only one color. Prove for n = 1: A bag with one ball clearly has balls of only one color... Find the error in the proof.

• Contrapositive [Beth, 02/27/2002]
  Am I allowed to say my reason is the inverse?

• Where is the Arsenic? [Shae, 03/12/2002]
  You place six jars (right to left: coffee, arsenic, and sugar on the top shelf; snuff, tea, and salt on the bottom shelf)...