Basic Definitions of Set Theory Lecture 24 Section 5.1 Fri, Mar 2, 2007

The Universal Set Whenever we use sets, there must be a universal set U which contains all elements under consideration. Typical examples are U = R and U = N. Without a universal set, taking complements of set is problematic.

Set Operations Let A and B be set. Define the intersection of A and B to be A  B = {x  U | x  A and x  B}. Define the union of A and B to be A  B = {x  U | x  A or x  B}. Define the complement of A to be A c = {x  U | x  A}.

Sets and Boolean Operators A set may be represented as a sequence of true and false values. Let the universal set be U = {a 1, a 2, a 3, …}. Then the set A = {a 1, a 3, …} could be represented as {T, F, T, …} or as {1, 0, 1, …}.

Sets and Boolean Operators What boolean operations correspond to the set operations of union, intersection, and complementation?

Set Differences Define the difference A minus B to be A – B = {x  U | x  A and x  B}. Define the symmetric difference of A and B to be A  B = (A – B)  (B – A).

Sets and Boolean Operators What boolean operations correspond to the set operations of difference and symmetric difference?

Subsets A is a subset of B, written A  B, if  x  A, x  B. A equals B, written A = B, if  x  A, x  B and  x  B, x  A. A is a proper subset of B, written A  B, if  x  A, x  B and  x  B, x  A.

Sets and Boolean Operators Is there a boolean operator that corresponds to the subset relation? That is, an operation * on boolean variables such that A*B is true if and only if A  B?

Sets Defined by a Predicate Let P(x) be a predicate. Define a set A = {x  U | P(x)}. For any x  U, If P(x) is true, then x  A. If P(x) is false, then x  A. A is the truth set of P(x).

Sets Defined by a Predicate Two special cases. What predicate defines the universal set? What predicate defines the empty set?

Intersection and Union Let P(x) and Q(x) be predicates and define A = {x  U | P(x)}. B = {x  U | Q(x)}. Then the intersection of A and B is A  B = {x  U | P(x)  Q(x)}. The union of A and B is A  B = {x  U | P(x)  Q(x)}.

Complements and Differences The complement of A is A c = {x  U |  P(x)}. The difference A minus B is A – B = {x  U | P(x)   Q(x)}. The symmetric difference of A and B is A  B = {x  U | P(x)  Q(x)}.

Subsets A is a subset of B if  x  U, P(x)  Q(x), or  x  A, Q(x). A equals B if  x  U, P(x)  Q(x), or  x  A, Q(x) and  x  B, P(x). A is a proper subset of B if  x  A, Q(x) and  x  B,  P(x).