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MA100 - BAUMAN - MA100 Fall 2010 PART 1 Precalculus[1 The...

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MA100, Fall 2010, PART 1: Precalculus [1] The Set of Real Numbers 1. Sets in general. Notations A set is a collection of objects, called the elements or members of the set. The elements of a set are listed between braces (curly brackets). The elements of a set are listed only once and the order in which these elements are listed is irrelevant. Example: { red, blue, green } is a set formed by the colors red, blue and green. Example: { 22 , 23 , 24 , 25 } is a set formed by the numbers 22 , 23 , 24 , 25. Notation : Sets are usually denoted by capital letters. Example: A = { b, d, a, f, g, h } is a set denoted by A and it is formed by the letters b, d, a, f, g, h . Example: X = { 22 , 23 , 24 , 25 } is a set denoted by X and it is formed by the numbers 22 , 23 , 24 , 25. Notation : The elements of a set can be either listed one by one, or described by a property that characterizes them. Example: The set X = { 22 , 23 , 24 , 25 } can be equivalently written as X = { the integers x with the property that x is greater than 21 and x is less than 26 } or X = { the integers x such that 21 < x < 26 } or X = { the integers x : 21 < x < 26 } Note that above, “ : ” is a short-hand notation for the words ”such that” . (Recall that , “21 < x ” is a short-hand notation for all numbers x that are greater than 21. Similarly, x < 26 is a short-hand notation for all numbers x that are less than 26.) Notation : The membership of an element to a set is denoted by the symbol . The non- membership is denoted by / . Example: Consider the set A = { b, d, a, f, g, h } . g A which reads “ g is an element of the set A ” or “ g is in A 2 / A which reads “ 2 is not element of the set A ” or “ 2 is not in A m / A which reads “ m is not element of the set A ” or “ m is not in A Definition : A set that contains no element is called the empty or the void set and it is denoted by the greek letter φ (which is read “phi”). Definition : Two sets are equal (or the same) if they consists of the same elements. In this case, we use the symbol “=.” If two sets do not contain the same elements, then they are not equal and we denote this by = . Example: Consider the sets A = { b, d, a, f, g, h } and B = { b, d, a, f, g, h } . Obviously, A = B. 1
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Example: Consider the sets A = { b, d, a, f, g, h } and B = { a, b, d, f, g, h } . Here also A = B, since A and B contain the same elements (recall that, when listing the elements of a set, the order doesn’t matter). Example: Consider the sets A = { b, d, a, f, g, h } and C = { b, d, f, g, h, 7 } . Then A = C. Definition : Let A and B be two sets. If all the elements of A are also elements of B , then we say that A is a subset of B and use the notation A B. If A B but A = B , then A is a proper subset of B , denoted A B. Example: Consider the sets A = { b, d } and B = { b, d, g, h } . Then A B. Also, since A = B, we can be more precise and write A B. Example: Consider the sets X = { the integers x : x < 11 } and Y = { the integers x : x < 15 } .
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