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Unformatted text preview: 1 Introduction to Set Theory Prepared by: Kendra Cooper Contents • Cantor’s Naïve Set Theory Introduction Set Operations Laws of Set Theory (Set Identities) Examples Using the Laws of Set Theory Problems Cantor’s Naïve Set Theory Introduction Definition Cantor was the first mathematician to propose a set theory (1874) • At the time, it was considered very controversial! A set is an unordered collection of objects. • This simple, intuitive definition of a set can lead to logical inconsistencies (or paradoxes, like Russel's Paradox) if we don't restrict the types of objects permitted in a set. • The objects in a set are called its elements or members. We say that a set contains its members. Textual Notations There are several ways to describe a set's members. • The first way is to list all members of a set between braces and denote the set with an uppercase letter. A ={1,2,4,apple} • If a set has infinite members or a very large number of members, the second way to list the member is to use ellipses. ,...} 2 , 1 , , 1 , 2 {..., = B • The third way to describe a set is to use the set builder, or construction, notation. C = { i 2  i is a nonnegative integer} 2 The bar notation reads "such that" To express membership in a set we use the notation A ∈ 4 . • This is a proposition that is true if 4 is a member of A. In the set A described above, this is true. Venn Diagrams Another way to describe sets and to illustrate its relationship to other sets is to use a Venn Diagram. • The universe is represented by a rectangle or square. • Circles are drawn inside the box to illustrate the sets and the relationships between sets. • Particular elements may be represented using points with a label. Let A= {1,2,3}, B = {1,2,3,17,19} The Venn Diagram is: Venn Diagrams are very helpful, but do not constitute a proof. Subsets A subset is an important relationship between sets. Set A is said to be a subset of B iff every element of A is also an element of B. B A ⊆ iff B x A x x ∈ → ∈ ∀ ( ) What is the domain of discourse for the universal quantifier? • In any given context there is a special set called the universe or universal set denoted by U such that every set in the given context is a subset of the universe. • In other words, the U contains all possible elements in our universe. 3 Set Equality Two sets are equal iff the first set is a subset of the second set and the second set is a subset of the first set. Theorem: ) ( S T T S T S ⊆ ∧ ⊆ ↔ = To prove the biconditional is a tautology we have to prove 2 implications are true. We'll use a direct proof. Proof: Let T y S x ∈ ∈ , . x and y are arbitrarily chosen elements....
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 Spring '09
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