Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U C X which contains only one of those points, if and only if Np # Ng.

Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.

Label each of the following statements as either true or false. Every mapping on a nonempty set A is a relation.

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

Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).

[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]

True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.

Label each of the following statements as either true or false. 2. for all nonempty sets A and B.