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Unformatted text preview: Sets, Real Numbers and Intervals Sets and set operations A set is any welldefined collection of (mathematical) objects. A set can be specified by listing its elements , or members, within curly brackets { } . . . . . = A { } , , , 1 2 3 4 and = B { } , , , , 4 2 0 2 4 are two finite sets of integers or "whole numbers". The set of all integers is the infinite set = Z { } , , , , , , , , , , , , . . . .  5 4 3 2 1 0 1 2 3 4 5 . . . . , where . . . . indicates that the listing continues forever. The order in which the elements of a set are listed is immaterial. Thus the set A above could equally well be specified by the equation = A { } , , , 3 1 4 2 . The important notion behind the concept of a set is membership . A set is determined only by knowing precisely which objects belong to the set. We write, 3 ε Z , or ∈ 3 Z , to mean 3 "is a member of", or "is an element of", the set of integers Z. A set can be specified by means of a general description of its elements, which is sufficiently precise to determine exactly which objects belong to the set. The sets A and B above could be defined as follows, where the symbol "  " should be read as "such that". = A { } , n  n ε Z and ≤ 1 n ≤ n 4 , that is, A is the set of all integers which are greater than or equal to 1 and less than or equal to 4. = B { } , 2 n  n ε Z and ≤  2 n ≤ n 2 that is, B is the set of all integers of the form 2 n , where n is greater than or equal to  2 and less than or equal to 2. These are the even integers between  4 and 4 inclusive. The set of all even integers is = E { } 2 n  n ε Z = { } , , , , , , , , , , , , . . . .  10 8 6 4 2 0 2 4 6 8 10 . . . . , and the set of all odd integers is = O { } 2 n 1  n ε Z = { } , , , , , , , , , , , , . . . .  9 7 5 3 1 1 3 5 7 9 11 . . . . . The set of perfect squares of integers is = S { } and n 2  n ε Z ≤ n = { } , , , , , , , 0 1 4 9 16 25 36 . . . . . The intersection of two sets Given any two sets A and B , their intersection , denoted by ∩ A B , is the set of all elements which belong to both A and B , that is, it is the set of elements common to the two sets A an B . = ∩ A B { } and x  x ε A x ε B ________________ For example, if = A { } , , , 1 2 3 4 and = B { } , , , , 4 2 0 2 4 then = ∩ A B { } , 2 4 . If , E O and S are the sets of even integers, odd integers and squares of integers, as above, then = ∩ E S { } , , , , 0 4 16 36 . . . . = { } and 4 n 2  n ε Z ≤ n , and = ∩ O S { } , , , , 1 9 25 49 . . . . = { } and ( ) 2 n 1 2  n ε Z ≤ n . Also, since E and O have no elements in common, = ∩ E S { }, the empty set . The empty set is the set which contains no elements....
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 Spring '08
 Uri
 Real Numbers, Sets, Peter Stone, 16 digits, 24 digits, Malaspina UC, 88 digits

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