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# Lecture_9 - MA1100 Lecture 9 Sets Set Notations Set...

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1 MA1100 Lecture 9 Sets Set Notations Set Relations Set Operations

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MA1100 Lecture 9 2 Sets A set is a well defined collection of objects The set of all positive integers Z + The set of all rational numbers Q The set of all integer solutions of x 2 = 9 The set of all rational numbers of the form 1/n where n is a positive integer The set of all real numbers R The set of all positive real numbers less than 10 Example
MA1100 Lecture 9 3 Elements of a Set Let A denote a set. if y is an object that belongs to this set A, we say y is an element of A and write y œ A . If y is not an element of A, we write y A Example 1.5 œ Q 2 Q A set with finitely many elements is called a finite set . The number of elements in a finite set A is called the cardinality of A and is denoted by |A|. Example A = {a, b, c} |A| = 3

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MA1100 Lecture 9 4 A strange “set” Let T be the “set” containing every set A such that A does not belong to itself We can use set to refer to almost any collection of objects: • The set of all animals • The set of all NUS students Let S be the “set” of all sets. Then S œ S .
MA1100 Lecture 9 5 Roster Method Roster method is a way to specify elements of a set by listing . This method works for small finite sets sets with elements having a fixed pattern The set of all integer solutions of x 2 = 9 The set of all positive integers less than 10 {-3, 3} {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, …, 9}

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MA1100 Lecture 9 6 Roster Method The set of all even integers {…, -6, -4, -2, 0, 2, 4, 6, …} Infinite set with “ pattern Roster method is a way to specify elements of a set by listing . L 4 1 3 1 2 1 1 1 , , , , The set of all rational numbers of the form 1/n where n is a positive integer
MA1100 Lecture 9 7 Set Builder Notation The set of all positive real numbers less than 10 We can describe the set using set builder notation { x œ R | 0 < x < 10 } underlying set underlying condition If we denote the set of positive real numbers as R + we can also write the set builder notation as { x œ R + | x < 10 } Not all sets can be described using roster method:

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MA1100 Lecture 9 8 Set Builder Notation { x œ U | P(x) } General form U : underlying set for the variable x P(x): underlying condition in terms of x that describe the elements
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