Lec2.1-2

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L2.1-2 Lecture Notes: Set theory A set is a collection of objects from a universe. The objects are called members or elements . They are not in any order, despite how they may be written, nor may sets contain duplicate members. Sets may be denoted in several ways: (1). By enumeration, if finite. E.g. {1,2,3} (Same as {2,1,3} — same as {1,1,2,3}) (2). By enumeration with dots of ellipses, where obvious. {2,4,6 . .} (3). By use of a propositional function: {x | x>3} (Need to know universe!) Or in general, {x | P(x)} or “the set of all elements that make P true.” Here are some facts and definitions about sets: As always, there is U, the universe of discourse, lurking in the background. Much of what we do with sets involves logic, so the careful reader may want to review or have close at hand our notes on logic. Define: x S to mean “x is an element of S.” Note that x S is a proposition. We also define x S to mean ¬ (x S) or “x is not an element of S.” Define |S| as “cardinality of S” or the number of elements in S. Define: S T as S is a subset of T iff 2200 x(x S x T) STOP: You should read the logic as smoothly as if it were written by J.K. Rowling. If you cannot, go back and review logic! Define: S=T iff S and T contain the same elements, or iff 2200 x(x S x T) Theorem: S=T iff S T T S (In words, two sets are equally if each is a subset of the other.) Proof: S=T iff 2200 x(x S x T) iff 2200 x[(x S x T) (x T x S)] iff 2200 x(x S x T) 2200 x(x T x S) iff S T T S QED Define: , or the null or empty set, as the set with no elements. That is, the following is a true proposition: 2200 x( ¬ (x ∈∅ )) or 2200 x(x ∉∅ ) 1

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Theorem: for any set S, ∅⊆ S Proof: ∅⊆ S iff 2200 x(x ∈∅ x S) Since x ∈∅ is false for all x, 2200 x(x ∈∅ x S) is true and this is sufficient to prove ∅⊆ S. Theorem: for any set S, S S. Proof: S S iff 2200 x(x S x S) Since P P is a tautology the quantified statement is true , and the theorem is proved. Sets may be members of sets. E.g. S = {1,2, {1,2}} is a set of cardinality 3, containing two numbers and a set as members. The most important set that contains sets as members is the power set of a set S, denoted P (S), which is the set of all subsets of S . E.g., Let S={a,b}. P (S) = P ({a,b}) = { , {a}, {b}, {a,b}} P ( ) = { } (The set containing as its only member the null set) P ( P ( )) = P { } = { , { }} P(P ( P ( ))) = P { , { }} = { , { }, {{ }}, { , { }}} STOP: Think about this! What is
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## Lec2.1-2 - &lt;?xml version=&quot;1.0&quot;...

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