lecture1

# lecture1 - Notes 1 for Honors Probability and Statistics...

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Unformatted text preview: Notes 1 for Honors Probability and Statistics Ernie Croot August 26, 2003 1 Set Theory 1.1 Basic Definitions In mathematics a set is a collection of elements or objects . We also allow S to have no objects, and we call this special kind of set the empty set , and denote it by ∅ . If S denotes a set, the symbol ∈ is used to express when an object is a member of S : The string of symbols “ a ∈ S ” is shorthand for the statement “ a is a member of the set S ”. Some common sets that you are no doubt familiar with include Z , the set of integers; N , the set of positive integers; Q , the set of rational numbers; R , the set of real numbers; and C , the set of complex numbers. We also have the notion of a subset and superset: Given sets A and B , if we have that every element a ∈ A is also an element of B – that is, a ∈ B – then we say that A is a subset of B , and we denote it by A ⊆ B . We say that the sets A and B are equal , and we denote it by A = B , if A ⊆ B and B ⊆ A ; also, if A is not equal to B , we write this as A 6 = B . If A is a subset of B and A 6 = B , then we say that A is properly contained in B or that A is a proper subset of B , and we write this as A ⊂ B . We also use the notation B ⊇ A and B ⊃ A , and it is hoped that you can deduce its meaning. One final thing to note is that the empty set is always a subset of A ; that is, ∅ ⊆ A . If A is not the empty set, then ∅ is a proper subset of A , which means we would use the notation ∅ ⊂ A . One common way to define a set is to list out the elements contained within it: For example, we could say that S = { 1 , 2 , 3 , 4 } ; that is, S is the set of all integers between 1 and 4, inclusive. For sets that have too many 1 elements to list out, but which can be algorithmically generated, one uses the notation S = { s : ... } , where the “...” identifies the properties satisfied by s . This notation for S is usually read out loud as “ S is the set of all elements s such that ...”. For example, if we let S = { n ∈ Z : n/ 2 ∈ Z } , one reads this as “ S is the set of integers n such that n/ 2 is also an integer”. It is obvious that this definition of S is a roundabout way of saying that S is the set of even integers. Sets do not have to be just numbers, but can be any collection of objects we desire to work with; for example, the set Z [ x ] usually means the set of polynomials in the variable x , so this set is all polynomials of the form a k x k + a k- 1 x k- 1 + ··· + a , where the a i ’s are integer and k can be any non- negative integer. Notice here that not every element of Z [ x ] is a number. There is a problem when one allows too much freedom for how one defines sets algorithmically, and this is perhaps best illustrated by something called Russell’s paradox, in honor of the famous British Philospher and mathemati- cian Bertrand Russell: First note that the elements of a set S can themselves be sets; for example the set...
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lecture1 - Notes 1 for Honors Probability and Statistics...

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