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Basic Definitions of Set Theory

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 .

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Basic Definitions of Set Theory

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

  2. 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.

  3. 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 Ac = {x U | xA}.

  4. Sets and Boolean Operators • A set may be represented as a sequence of true and false values. • Let the universal set be U = {a1, a2, a3, …}. • Then the set A = {a1, a3, …} could be represented as {T, F, T, …} or as {1, 0, 1, …}.

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

  6. Set Differences • Define the differenceAminusB 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).

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

  8. Subsets • A is a subset of B, written A B, if x  A, x  B. • AequalsB, 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.

  9. 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?

  10. 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).

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

  12. 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)}.

  13. Complements and Differences • The complement of A is Ac = {x U | P(x)}. • The differenceAminusB 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)}.

  14. Subsets • A is a subset of B if • xU, P(x) Q(x), or • xA, Q(x). • AequalsB 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).

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