Let B be the collection of all Borel sets in R.. For any such Borel set U, define B U as the collection of all intersections between U and the members of B, that is B U= {BU: Be B}. Show that B U is a o-algebra in U.

Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andB

[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]

In Exercises 13-24, prove the statements concerning the relation on the set of all integers. 19. If and, then.