Boolean rings and Boolean algebras

  • Context: Graduate 
  • Thread starter Thread starter quasar987
  • Start date Start date
  • Tags Tags
    Rings
Click For Summary
SUMMARY

This discussion clarifies the relationship between Boolean rings and Boolean algebras, specifically how a Boolean algebra is derived from a Boolean ring using the operations defined as xANDy=xy, xORy=x+y+xy, and xNOT=1+x. The participant raises a critical point regarding the involution property of xNOT, arguing that (xNOT)NOT does not equal x, leading to the conclusion that xNOT should be defined as -x to satisfy this property. The discussion also touches on the idempotent nature of Boolean rings, where x+x=0, reinforcing the unique characteristics of Boolean algebra.

PREREQUISITES
  • Understanding of Boolean rings and their operations
  • Familiarity with Boolean algebra concepts
  • Knowledge of idempotent rings and their properties
  • Basic grasp of algebraic structures in mathematics
NEXT STEPS
  • Study the properties of Boolean rings in detail
  • Explore the implications of involution in algebraic structures
  • Investigate idempotent rings and their applications
  • Learn about the relationship between Boolean algebras and other algebraic systems
USEFUL FOR

Mathematicians, computer scientists, and students studying algebraic structures, particularly those interested in the foundations of Boolean algebra and its applications in logic and computation.

quasar987
Science Advisor
Homework Helper
Gold Member
Messages
4,796
Reaction score
32
My professor wrote that we get a Boolean algebra from a Boolean ring (R,+,-,.,0,1) by setting xANDy=xy, xORy=x+y+xy and xNOT=1+x.

But it seems to me that xNOT is not an involution. I.e., (xNOT)NOT = 1+(1+x), which is not x.

(xNOT=-x would do the trick though)
 
Physics news on Phys.org
It seems to me that

1+(1+x) = x

For several reasons. What else would it be equal to? Doesn't 1 + 1 = 0?
 
I forgot about that. In an idempotent ring, x+x=0.
 

Similar threads

Undergrad What Are the Axioms of Fuzzy Logic and How Do They Extend Boolean Algebra?
  • · Replies 7 ·
Replies
7
Views
2K
High School Traditional logic and its usefulness in the past
  • · Replies 21 ·
Replies
21
Views
3K
Graduate How Can Boolean Polynomials Determine Consistency in Propositional Formulas?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How to determine if a set is a semiring or a ring?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Proving Boolean Lattice Complementarity in [a,b]
  • · Replies 3 ·
Replies
3
Views
2K
MHB Distribution, expected value, variance, covariance and correlation
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Boolean Algebra: Proving A'''' = A Using Complement and Binary Addition
  • · Replies 5 ·
Replies
5
Views
3K
MHB Calculating Conditional Probabilities in Boolean Algebra
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Boolean Rings: System of Switches.
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
1K