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Dr. Math FAQ:
Browse High School Logic
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
- Finding the Power Set of a Power Set [02/17/2005]
What is "the second power set", or the power set of the power set of a
set, say set <1,2,3>?
- Five-Set Venn Diagram? [11/25/2001]
What does a five-set Venn diagram look like?
- Football Logic: Who Defeated Whom...? [8/17/1996]
Four teams were brown, blue, red, and purple; the competing captains were
Albie, Barry, Bill, and Ben... who defeated whom?
- The Four Doors of Xanth [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.
- Fuzzy Logic [04/26/1997]
What is fuzzy logic?
- Game Theory and Payoff Matrices [11/14/1998]
Can you give me a good introduction to game theory? What is a payoff
- Gauss' Method for Solving Equations [12/11/1998]
How do you use Gauss' method and matrices to solve systems of equations?
Why does this method work?
- A Geometry and a Logic Problem [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.
- Geometry Puzzles [12/18/1995]
A student asks Dr. Math for help in finding the correct combinations of
numbers to solve two puzzles.
- Godel's Incompleteness Theorem [01/18/1999]
What does Goedel prove in his incompleteness theorem?
- Goedel's Incompleteness Theorem [08/08/1997]
How did Goedel prove that any nontrivial logical system cannot be proven
to be inconsistent?
- History of the Symbol for "Therefore" [11/14/2005]
Why is it that the symbol for "therefore" is a centered dot with two
lower dots? Where did that symbol come from?
- How Can a Set Be Empty? [09/29/2003]
Why is the empty or null set called a set when it has no elements?
- How Many are in the Group? [10/17/1996]
Everyone in the group had been to at least one of the parks...
- How Many Pieces of Candy in Each Jar? [09/30/1997]
At the annual Cumberland County fair, one of the more popular booths is
the Candy Contest...
- A Hundred Thousand Switches, One Defect: How Few Tests? [04/19/2011]
Seeking to minimize the binary tests necessary to identify a manufacturing defect, a
student imagines representing the problem as a two-dimensional array, and
pursuing that strategy into higher dimensions. Doctor Anthony introduces a bit-wise
approach for determining the least number, then proceeds through a smaller example
to demonstrate the general method for solving such problems of efficiently identifying
the one bad apple.
- If It Doesn't Follow a Form, Don't Draw a Conclusion [09/22/2011]
A student of syllogisms and detachments struggles with statements that do not map to
those forms. By rhetorically extending her examples, Doctor Peterson reveals that
sometimes, the only thing you can conclude is "No conclusion possible."
- If P then Q [08/29/1997]
I don't understand how if p is false then regardless of q the statement
- Implications in Logic [10/29/2002]
I don't understand the first four rules to do with implications:
Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive
- Inclusion-Exclusion Principle [09/03/2002]
In a survey of 100 people, 85 like calypso and 93 like pan. Calculate
the number of people who like both calyso and pan.
- The Indeterminate Nature of 0/0 [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?
- Indirect Proof of Parallel Lines [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...
- Indirect Proofs [09/21/1997]
If Clark is a mathemagician, then Lois is his assistant...
- Induction Proof with Inequalities [07/03/2001]
Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative
- Inductive Misunderstanding [04/26/2012]
One explanation of mathematical induction leads a student to question the method's
validity; a second makes him doubt its purpose. Doctor Rick re-lays the groundwork
for logic's "infinite staircase."
- Inductive vs. Deductive Reasoning [07/24/2001]
Can you explain the difference between inductive and deductive reasoning?
- Infinite and Transfinite Numbers [5/28/1996]
Can anyone explain to me, in a simple way, what transfinite numbers are
and how they're different from infinite numbers?
- 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.
- Interchanging Variables in Logic [04/10/2003]
What are the four rules of logic? Can the variables (p and q) be
interchanged? Why or why not?
- Interpreting Intersections [07/09/2002]
Out of 100 people, 10 drink Coke, 15 drink Sprite, and 20 drink Pepsi.
How many drink both Coke and Pepsi at the same time? Coke and Sprite?
Pepsi and Sprite? All three?
- Introduction to Logic and Truth Tables [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?
- Invalid Logic Argument [9/9/1996]
If I want the result to be true but the premise is false, the argument
will be invalid...
- Is Henry Guilty? (Geometry Puzzle) [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...
- Is It Possible That x/0 is Not Really Undefined? [06/14/2004]
Is it possible that 1/0 is not really undefined, but is in fact the
key to a whole new class of numbers, much as the imaginary unit i led
to the development of complex numbers?
- John Venn and Venn Diagrams [09/04/1998]
Can you give me some information on John Venn and the origin of Venn
- Karnaugh Maps [05/07/2000]
What are Karnaugh maps? How are they used?
- Knowing People at a Party [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...
- Largest x, x^2 less than 2 [10/23/1999]
Prove that there is no largest real number x, such that x^2 is less than
2. (Use indirect proof.)
- Lewis Carroll's Logic Problems [01/15/1997]
Where can I find out more about Lewis Carroll's logic problems?