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Browse College Logic and Set Theory

Stars indicate particularly interesting answers or good places to begin browsing.



About Fuzzy Logic [05/06/2003]
What is fuzzy logic? What's the difference between fuzzy logic and Boolean logic?

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

Lines, Points, and Infinities [09/01/2001]
What is the cardinality of the set of real numbers between 0 and 1? Is this cardinality less than, greater than, or equal to the cardinality of real numbers between 0 and 2?

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

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...

The Truth-teller, the Liar, and Ambiguous [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.

Advanced Topics in Symbolic Logic [10/28/2002]
I cannot find any unsolved proofs that I can just solve for fun.

Are Different Proofs of a Theorem Really the Same? [07/05/2006]
If you have a mathematical system with several axioms (call them A, B, C, D, E and F), is it possible to have two proofs of a theorem in this system where one proof uses only axioms A, B, and C and the other proof uses only axioms D, E, and F? In other words is it possible for two proofs to use no common axioms? Or are all proofs of that theorem really based on the same set of axioms?

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

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

Building Sets [05/26/2002]
Is 5 part of the set {x:x is a multiple of 7 and 5 < x < 56}?

Can Rewriting P -> Q as ~Q -> ~P Lead to a False Conclusion? [01/21/2006]
An interesting logic puzzle about determining a birthday leads to a discussion about interpretation, logic, and seeming confusion when one of the logic statements is rewritten in a different but equivalent form.

Cardinality between Open and Closed Sets [09/20/2001]
I would like to know how to prove that the sets (0,1) and [0,1] have the same cardinality.

Cardinality of Euclidean Space [09/13/2005]
What is the cardinal number of a n-dimensional Euclidean space R^n where n tends to aleph_0, aleph_1, aleph_2, and so on?

Claim the Last Flag [10/25/2002]
Two teams face 21 flags. Teams take turns choosing 1, 2, or 3 flags at each turn. The team that can claim the last flag wins.

Closed Sets [02/27/1999]
Is a union of finite number of closed sets and the intersection of any number of closed sets closed?

Comparing Size of Infinite Subset to Parent Infinity Subset [12/16/2001]
If one considers the single, original set of all positive integers the "physiological" state and wants to compare the size of the positive even integers to the total positive integers, doesn't the splitting out of the evens into a separate subset constitute an experimental artifact that doesn't accurately reflect the original state?

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

Constructing Truth Tables [02/17/2006]
Construct a truth table for the statement ~q v (p^r).

Constructing Truth Tables for More Than Three Statements [03/07/2004]
I know how to do truth tables for up to three statements. How do I continue with a fourth or fifth statement such as (p ^ q) -> (rvs)?

The Continuum Hypothesis [5/24/1995]
What is it the continuum theory, and has it been resolved?

Contrapositives and Monotonic Functions [03/17/2003]
Defining 'monotonic' (increasing or decreasing functions) and 'contrapositive'.

Countability of Primes and Composites [05/18/2002]
If the union of two sets is countable, can either of the sets be uncountable?

Countability of Sets [09/23/2004]
Let A be the set of all sequences of the form (d1, d2, 23,...) where di = 0 or 1 and there are only a finite number of 1's in the sequence. Prove that A is countable.

Countability of Sets [10/20/2004]
The set Q of all rational numbers and the set N2 of all ordered pairs of natural numbers are countable. Show that the following sets are countable: (a) Q^2; (b) The set of all subsets of N of size 2; (c) The set of all subsets of N of size n, for some fixed (but arbitrary) n E N; (d) The set of all finite subsets of N.

Countable and Uncountable Infinities [08/13/2009]
Could you please explain the terms 'countable' and 'uncountable' infinity? Are uncountable infinities greater than countable?

Countable Sets and Measure Zero [05/12/2001]
How would you prove that if a set S is countable, then S has measure zero?

Defining Multiplication [08/16/2002]
What is the distinction between 3*4 and 4*3?

DeMorgan's Laws and Distribution Rules [05/06/2003]
How can I show the reduction of the following problem without using tables?

Distinguishing between Two Random Sequences [05/09/2008]
Is there any efficient way, such as a polynomial-time algorithm in L, to distinguish between a distribution which takes L elements at random in [1, N] and one which takes L elements at random in [1, N], then picks a random v in [1, L] and replaces x_v with a random element in [1, N/2]?

Divisor Proof with Contrapositive [09/16/2002]
I have been trying to prove that if n^2 divides m^2, then n divides m.

Equivalence Relations [12/10/2001]
Let X={1,2,3,4,5}, Y={3,4}. Define a relation R on the power set of X by A R B if A U Y = B U Y. Prove that R is an equivalence relation. What is the equivalence class of {1, 2}? How many equivalence classes are there?

Equivalence Relations [02/10/2003]
For integers m and n, define m~n if n|m^k and m|n^k for some positive integers j and k.

Explain Supremum [02/02/1998]
Can you please explain, perhaps with an example, the concept of "supremum"?

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>?

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

Groups and Subgroups [10/20/2001]
Show that a subset of a group is a subgroup. Understanding how properties distinguish elements.

Hyperbolic Geometry [03/24/2003]
Explain this assumption: Assuming that Euclidean geometry is consistent, had any of the failed attempts to prove Euclid's 5th Postulate from the other axioms succeeded, they would have actually completely destroyed Euclidean geometry as a consistent body of thought.

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

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?

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