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ClockNumbersAndSets

# ClockNumbersAndSets - 1 Sets and Searching 1.1 Introduction...

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1 Sets and Searching 1.1 Introduction to Sets A set is a collection of objects. We say that the objects in a set are “elements” of the set. Or we may say that they “belong to the set” or we may say that they are “in the set” or we may say that they are “members of the set”. 1.1.1 Representing a set by listing its elements We represent it using { }. This is read as “the set containing. ..” and you complete the phrase by looking at what is between the braces – the elements of the set. Here are some sets. {a, A, b} {New Jersey, Connecticut, Hawaii} {Gov. Christie, President McCormick, Bugs Bunny} The elements do not have to be of the same type (such as, for example, letters, or states, or well known names). So this is a possible set: {Bugs bunny, New Jersey, A } All of these sets have the same size. Written as |{. ..}|. It is useful to give sets concise representations so we can talk about them. We can do this using capital letters (that's the usual way). S 1 {a, A, b} S 2 S 3 S 4 {Bugs bunny, New Jersey, A } 1.1.2 Notation for the Size of a Set So we can write || S 4 3 And so forth 1.1.3 Notation for an element or member of a set We need a notation to say that one of the objects is a member of a set, and we do that with something

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that looks kind of like the Greek letter epsilon: . For example, we write AS 4 ; or, President McCormick S 3 and so forth. We can also represent sets by giving a defining rule for the members of the sets. Let's do this using numbers. And to keep things easy to handle let's start with the “Clock Numbers” 1.1.4 The Set of Clock Numbers C={1,2,3,4,5,6,7,8,9,10,11,12}. (we are calling this set “C” for now. Later we will call other sets “C”, so this is not a permanent name. 1.1.5 Defining a set by the properties of its elements Some of the clock numbers are divisible by 3. We can define a specific set using that property: { : } C x x C and x is divisible by  3 3 This is pronounced: “C sub 3 is the set of all objects, x, such that x is divisible by 3”. Class calisthenics: How many elements are there in each of these sets: ? ? ? ? ? ? C C C C C C 1 2 3 7 12 13 1.1.6 The Empty Set What about that last one. It has no elements. We call it “ the empty set”. We do we say “the”? Because no one has any way to telling one empty set such as C 13 , from an apparently different empty set such as C 31 . This idea turns out to be very useful, and mathematicians have a specific symbol for the empty set: . So, mathematically, we can say that “none of the clock numbers are divisible by 13”, by
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ClockNumbersAndSets - 1 Sets and Searching 1.1 Introduction...

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