1. “It is impossible to define every concept.” For example a “set” can not be defined. But Here are a list of things we shall simply assume about sets. A set S is made up of elements, and if a is one of these elements, we shall denote this fact by There is exactly one set with no elements. It is the empty set and is denoted by SECTION 0 SETS AND RELATIONS

2. Two Ways to Describe a Set List the elements: –enclose the elements, separated by commas in braces. –Ex: {1, 3, 5}. Give a characterizing property P(x) of the elements x: –“Set-builder notation” {x | P(x)} –“the set of all x such that the statement P(x) about x is true.” –Ex: {2, 4, 6} = {x| x is an odd whole positive number <8}

3. Well Defined A set is well defined if S is a set and a is some object, then either a is definitely in S, denoted by, or a is definitely not in S, denoted by Q: Decide which of the following is well defined:

4. Definitions Definition: A set B is a subset of a set A ( ), if every element of B is in A. Note: for any set A, A and are both subsets of A. Definition If A is any set, then A is the improper subset of A. Any other subset of A is a proper subset of A. Q: Let S = {1, 2, 3}, list all eight (proper) subsets of S.

5. Cartesian Product Definition: Let A and B be sets. The set is the Cartesian product of A and B. Q: If A = {1, 2} and B = {3, 4}, then what is A X B?

6. Notations Z is the set of all integers Q is the set of all rational numbers R is the set of all real numbers C is the set of all complex numbers Z +, Q +, and R + are the sets of positive members of Z, Q, and R, respectively. Z *, Q *, R * and C * are the sets of nonzero members of Z, Q, R, and C, respectively.

7. Relations Between Sets Definition: A relation between sets A and B is a subset R of A X B. We read R as “a is related to b” and write a R b. Example: Equality Relation = defined on a set S is the subset Function Relation f(x) = x 3 for all x  R, is the subset of R X R

8. Function A function  mapping X into Y is a relation between X and Y with the property that each x  X appears as the first member of exactly one ordered pair (x, y) in . Such a function is also called a map or mapping of X into Y. We write  : X  Y and express (x, y)   by  (x)=y. The domain of  is the set X and the se Y is the codomain of . The range of  is  [X] = {  (x)|x  X}.

9. Function cont. A function  : X  Y is one to one if  (x 1 )=  (x 2 ) only when x 1 =x 2. The function  is onto Y if the range of  is Y. Q: Exercise 12.

10. Cardinality The number of elements in a set X is the cardinality of X and is often denoted by |X|. Example: |{1, 2, 4, 6}|=4 Two sets X and Y have the same cardinality if there exists a one-to-one function mapping X onto Y, that is, if there exists a one-to-one correspondence between X and Y.

11. Partitions Sets are disjoint if no two of them have ay any element in common. Definition A partition of a set S is a collection of nonempty subsets of S such that every element of S is in exactly one of the subsets. The subsets are the cells of the partition. Q: exercise 23-27 Each partition of a set S yields a relation R on S in a natural way: namely, for x, y  S, let x R y if and only if x and y are in the same cell of the partition.

12. Equivalence Relation An equivalence relation R on a set S one that satisfies these three properties for all x, y, z  S. 1.(Reflexive) x R x. 2.(Symmetric) If x R y, then y R x. 3.(Transitive) If x R y and y R x, then x R z. Q: Let a relation R on the set Z be defined by n R m if and only if nm  >=0. Determine if R is an equivalence relation.

13. Equivalence Relations and Partitions Theorem Let S be a nonempty set and let  be an equivalence relation on S. Then  yields a partition of S, where Also, each partition of S gives rise to an equivalence relation  on S where a  b if and only if a and b are in the same cell of the partition. Q: exercise 29-32