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Dr. Math FAQ:
Browse High School Logic
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Selected answers to common questions:
- 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 Equivalents [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)).
- Logical Fallacies [01/29/2001]
Don't fallacious arguments (as in the argumentum ad ignorantiam)
- 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?
- The Logic behind Conditional Statements [11/28/2007]
I understand the subset explanation of why the conditional logic
statement 'If false then true/false' is always considered true. But
what is the logic behind it?
- Logic, Groups, and Identities [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?
- Logic Laws [03/04/2003]
I do not understand the laws of inference, simplification, disjunctive
inference, and disjunctive addition.
- Logic - Liars & Truthtellers (What Question Does She Ask?) [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.
- Logic Statement False Implies True [02/06/2008]
The logic statement A->B is considered true if A is false and B is
true. How can a false imply a true? What's the thinking behind that
statement and can you give a good example of how it works?
- Making Heads or Tails of Binary Proofs [03/17/2011]
Can 5 tails-up coins become all heads-up by turning them over 2 at a time? If the total number of coins and turned coins changes, what patterns emerge? Doctor Tom introduces mathematical invariants and monovariants to confirm a student has understood that these questions seek proofs -- and that she has started down the right path of proving the possibility (or impossibility) of certain states.
- Match Couples and Parties [03/28/2001]
Read the clues given, and match everything up.
- Mathematical Induction [09/07/1998]
What is mathematical induction? Can you give an example of the ideas of
- Mathematical Induction [07/01/1998]
Proof by induction does not prove anything, because in the inductive
step, one makes the assumption that P(k) is true...
- Mathematical Logic [02/09/2001]
Assumptions, rules, contradictions, and a derivation.
- Mathematics, Logic, and Intuition [05/27/2003]
How is math related to logic and intuition?
- Math Logic [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?
- Math Logic - Determining Truth [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.
- Math Symbol for IFF [06/18/2003]
Is there a mathematical symbol for the term 'if and only if'?
- 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?
- Minimal Weighings of Ten Coins to Identify the Two Counterfeits [04/18/2010]
At least how many balance scale weighings of ten coins do you need to determine the
two fakes? By applying combinatorics and keeping track of lower bounds, Doctor
Jacques provides a methodical approach.
- Modus Ponens [07/10/2001]
A man born in 1806 is x years old at the year x squared. Solve for x.
- Monty Hall Logic [03/09/2001]
Are there in fact four options? Aren't there three choice points, not
- Monty Hall Strikes Again [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?
- 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.
- The Necessary Condition and the Sufficient One [10/21/2015]
What is the sufficient condition in p --> q? What is the necessary condition? Doctor
Peterson explains the logic, illustrating with the contrapositive as well as an example
from recent space exploration.
- Negating a Quantifier [03/20/2010]
What is the negation of "at least two"? Is it "none" or "at most two"?
Doctor Peterson responds by analyzing one case at a time as well as by
representing the proposition as an inequality.
- Negating Statements [10/27/1998]
What is negation? What is a statement? How do you negate a statement?
- Negation in Logic [8/3/1996]
What is the negation of "In every village, there is a person who knows
everybody else in that village"?
- 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?
- Open Sentence, Statement [09/18/2001]
What is an open sentence?
- 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...
- Paradox [05/07/2001]
What is a paradox?
- Paradox and Fallacy [01/25/2001]
What is the difference between paradox and fallacy in mathematics?
- Paradox of the Unexpected Exam [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?
- 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?