Math 113 Final Exam Solutions
... 1. a) (5 points) A ring R is called Boolean if a2 = a for all a ∈ R. Show that xy = −yx for all x, y ∈ R in a Boolean ring R. (Hint: consider (x + y)2 ) (x + y)2 = x2 + xy + yx + y2 . Since R is Boolean, we also have (x + y)2 = x + y as well as x2 = x, y2 = y. So we have x + y = (x + y)2 = x2 + xy + ...
... 1. a) (5 points) A ring R is called Boolean if a2 = a for all a ∈ R. Show that xy = −yx for all x, y ∈ R in a Boolean ring R. (Hint: consider (x + y)2 ) (x + y)2 = x2 + xy + yx + y2 . Since R is Boolean, we also have (x + y)2 = x + y as well as x2 = x, y2 = y. So we have x + y = (x + y)2 = x2 + xy + ...
Boolean rings and Boolean algebra
... The word ring as it is used measure theory corresponds to the notion of ring used elsewhere in mathematics, but I didn’t give the correct correspondence in lecture. I will do so now. A (commutative) ring is, by definition, a set with two commutative operations, addition and multiplication. The ring ...
... The word ring as it is used measure theory corresponds to the notion of ring used elsewhere in mathematics, but I didn’t give the correct correspondence in lecture. I will do so now. A (commutative) ring is, by definition, a set with two commutative operations, addition and multiplication. The ring ...
