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22 March 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction to Discrete Mathematics Sets Part1

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22 March 2009Instructor: Tasneem Darwish2 Outlines Sets and Membership. Subsets. Operations on Sets.

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22 March 2009Instructor: Tasneem Darwish3 a set is to be thought of as any collection of objects whatsoever. The objects can also be anything and they are called elements of the set. The elements contained in a given set need not have anything in common. we should be able to decide whether an element belongs to the set. Example 3.1: 1.A set could be defined to contain Picasso, the Eiffel Tower and the number π. 2.The set containing all the positive, even integers is clearly an infinite set. 3. Consider the ‘set’ containing the 10 best songs of all time. This is not allowed unless we give a precise definition of ‘best’. Sets and Membership

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22 March 2009Instructor: Tasneem Darwish4 upper-case letters are used to denote sets lower-case letters are used to to denote elements. The symbol ∈ denotes ‘belongs to’ or ‘is an element of’. The simplest set definition is by listing the elements enclosed between curly brackets or ‘braces’ { } D = { }, the empty set (or null set), which contains no elements. This set is denoted. Sets and Membership

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22 March 2009Instructor: Tasneem Darwish5 Listing the elements of a set is impractical except for small sets An alternative is to define the elements of a set by a property or predicate if P(x) is a propositional function, we can form the set whose elements are all those objects a (and only those) for which P(a) is a true proposition. A set defined in this way is denoted Example 3.2: 1)The set B could be defined as B = {n : n is an even, positive integer}, or B = {n : n = 2m, where m > 0 and m is an integer} 2)The set {1, 2} could alternatively be defined as {x : x2−3x +2= 0}. 3) The empty set can be defined in this way Sets and Membership

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22 March 2009Instructor: Tasneem Darwish6 Two sets are defined to be equal if and only if they contain the same elements Theses are equal sets: Sets with different definitions but same elements are equal Equality of Sets

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22 March 2009Instructor: Tasneem Darwish7 If A is a finite set its cardinality, |A|, is the number of (distinct) elements which it contains. If A has an infinite number of elements, we say it has infinite cardinality, and write |A| = ∞. Other notations commonly used for the cardinality of A are n(A), #(A) and Examples 3.3 | | = 0 since contains no elements. |{π, 2, Attila the Hun}| = 3. If X = {0, 1,..., n} then |X| = n + 1. |{2, 4, 6, 8,...}| = ∞. |{{1,2}}|= 1 also |{ }|=1 Set cardinality

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22 March 2009Instructor: Tasneem Darwish8 The set A is a subset of the set B, denoted A ⊆ B, if every element of A is also an element of B. Symbolically, A ⊆ B if ∀ x [x ∈ A → x ∈ B] is true. If A is a subset of B, we say that B is a superset of A, and write B ⊇ A Clearly every set B is a subset of itself, B ⊆ B. The notation A ⊂ B is used to denote ‘A is a proper subset of B’. Thus A ⊂ B if and only if A ⊆ B and A ≠ B. ⊆ A for every set A. The proposition x ∈ is false subsets

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22 March 2009Instructor: Tasneem Darwish9 Examples 3.5 1.{2, 4, 6,...} ⊆ {1, 2, 3,...} ⊆ {0, 1, 2,...}. Of course, we could have used the proper subset symbol ⊂ to link these three sets instead. 2.Let X = {1, {2, 3}}. Then {1} ⊆ X but {2, 3} is not a subset of X. {2, 3} is an element of X, so {{2, 3}} ⊆ X. To prove that two sets are equal, A = B, it is sufficient to show that each is a subset of the other, A ⊆ B and B ⊆ A subsets

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22 March 2009Instructor: Tasneem Darwish10 Examples 3.6 (1) Show that Let and We need to show that every element of A is an element of B The equation 2x 2 + 5x − 3 = 0 has solutions x = ½ and x = −3, so A = {1/2,−3}. When x = 1/2, 2x 2 + 7x + 2 = 1/2+ 7/2+ 2 = 6 = 3/x, so ½ ∈ B. When x = −3, 2x 2 + 7x + 2 = 18 − 21 + 2 = −1 = 3/x, so −3 ∈ B. Therefore every element of A is an element of B, so A ⊆ B. subsets

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22 March 2009Instructor: Tasneem Darwish11 Examples 3.6 (2) Let A = {{1}, {2}, {1, 2}} and let B be the set of all non-empty subsets of {1, 2}. Show that A = B. Solution B={{1},{2},{1,2}} Thus, A ⊆ B and B ⊆ A We conclude that A=B subsets

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22 March 2009Instructor: Tasneem Darwish12 The universal set is frequently denoted. The universal set is essentially the universe of discourse introduced in chapter 1. Some special sets of numbers which are frequently used as universal sets are the following: Also frequently used are and, the sets of positive integers, rational numbers and real numbers respectively. Note that is not equal to Universal sets

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22 March 2009Instructor: Tasneem Darwish13 sometimes we use and to denote the sets of even and odd integers Universal sets

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22 March 2009Instructor: Tasneem Darwish14

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