REMEMBER TO BRING AN 8’’x11”  BLUE EXAM BOOKLET FOR THE TEST

               KEY CONCEPTS AND MAIN DEFINITIONS:

 

       Singleton sets, Unordered pairs, Ordered pairs, Set of sets, Union of a set of sets, Intersection
       of a set of sets, Indexed family of sets (their unions and intersections), Relations (reflexive,
       symmetric, transitive, anti-relexive, anti-symmetric, asymmetric, and connected  relations),
       Functions (injective, surjective, and bijective functions), Composition of functions, Generalized
       Cartesian products, The Logical & Proper Axioms of Set Theory, Classes & proper classes,
       Class-form of the Separation & Replacement Axioms, Partial orders, Total (linear) orders,
       Well-founded partial orders, Well-orders, Initial segments,  Transitive sets, Ordinals, Successor
       & Limit ordinals, Ordinal addition, Ordinal multiplication, Ordinal exponentitaion, Properties of
       the Cumulative Hierarchy.   

      
              MAIN PROBLEM SOLVING TECHNIQUES:

      

       1.  Proving various identities or subset-relations involving sets and set operations .

2.   Determining whether or not a class is a set with or without Proper Axioms.
3.      Proving properties of relations and various kinds of functions.

4.   Translating statements into the Language of Set Theory (LOST) especially the
10 Proper Axioms of Set Theory.

5.   Proving various properties of partially ordered sets (posets), of totally ordered sets
(tosets, linearly ordered sets, losets), and of well-ordered sets (wosets).

6.      Proving properties of transitive sets and of ordinals.

       7.  Defining various operations by using transfinite recursion.
       8.      Proving various statements about ordinals by using transfinite induction.

        9.      Proving various statements about the cumulative hierarchy by using transfinite induction.

               MAIN THEOREMS

       1.      R = {x : neg(xEx)}, V = {x : x = x}, and S_n = {x : |x| = n }, n>0, are proper classes.

2.    If W and X are well-ordered sets, then W is isomorphic to an initial segement of X,
       X is isomorphic to an initial segment of W, or W is isomorphic to X.
3.      The Transfinite Induction Theorem.

4.   The Transfinite Recursion Theorem.

5.   The theorems on the properties of Ordinal Arithmetic.