Main text book How to download the main

Main text book

How to download the main text for your course? 1. www. google. com 2. In the search field write “free download discrete mathematics and its applications by kenneth h rosen 7 th edition. Pdf ”, like the following figure. 3. Then press enter key.

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Sets and Subsets Prepared by: Raja’a Masa’deh First semester,

What is the meaning of set? q It is common for sets to be denoted using uppercase letters. Lowercase letters are usually used to denote elements of sets. q There are several ways to describe a set: ü Roster method. ü Set builder notation.

ü Roster method: sets are written using curly brackets {, } with their elements listed in between. Example (1): The set V of all vowels in the English alphabet can be written as V = {a, e, i, o, u}. Example (2): The set O of odd positive integers less than 10 can be expressed by O = {1, 3, 5, 7, 9}. Example (3): All odd numbers ( positive and negative) O = {…, -7, -5, -3, -1, 1, 3, 5, 7, …}. Infinite Curly bracket, set bracket, brace Element Called ellipses means continue on…

A= {1, 3, 5, 7} B={1, 3, 5, 7, …} Finite set Infinite set Note: Sometimes the “…” can be used in the middle to save writing long lists. Example : The set of letters : {a, b, c, … , x, y, z}. Is it finite or infinite set? It is finite set because there are only 26 letters.

ü Set builder notation: it is another way to describe a set. We characterize all those elements in the set by stating the property or properties they must have to be members. Example : The set O of all odd positive integers less than 10 can be written as O = {x | x is an odd positive integer less than 10}. Or O = {x ∈ Z+ | x is odd and x < 10}.

Universal Set q The universal set U contains all the objects under consideration, is represented by a rectangle. Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set. Example : Draw a Venn diagram that represents V, the set of vowels in the English alphabet.

Subset q The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B. Example : A= {r, s, t, u, w} and B={s, u, w} A B B is a subset of A. U

Example : A= {r, s, t, u, w}, B={s, u, w} and C={n, o, p}. C ⊄ A and A ⊄ C C ⊄ B and B ⊄ C A C B

Some principles q ∈ means “belong to”. q Element “ a” is in a set A. Thus, we use the symbol “∈ “ to show it as ( a ∈ A). q ∉ means “not belong to”. q In case an element doesn’t belong to a set, use “ ∉“. Example : A= { a, b, c, d} c ∈ A. ( which means c belongs to A). p ∉ A. ( which means p doesn’t belong to A).

Subset VS Proper Subset q set A is a subset of a set B but that A ≠ B, we write A ⊂ B and say that A is a proper subset of B. For A ⊂ B to be true, it must be the case that A ⊆ B and there must exist an element x of B that is not an element of A. That is, A is a proper subset of B. q In other words, If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊈ A i. e. , A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B. Example : A= {a, b, c, d}, B={a, c, d}, C= {d, c, b, a} ü C ⊆ A because all elements in C are found in A. Moreover, they are equal sets. But C is not proper subset of A. ü B ⊂ A; B is proper subset of A because the set B has less elements than A.

q A is a subset of B if and only if every element of A is in B. q ⊆ means “ Subset ”. q ⊈ means “not subset”. Example : A= {1, 2, 3, 4, 5} B= { 1, 2, 3, 3, 4, 5, 5}. B ⊆A , which means B is subset of A. Question : A= {1, 3, 4} B= { 1, 2, 3, 4}. Is A a subset or proper subset of B?

Example : Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? and Is B a subset of A? A= { …, -8, -4, 0, 4, 8, …} The answer B= { …, -8, -6, -4, -2, 0, 2, 4, 6, 8, …} A ⊂ B but B ⊄A. Empty Set: q is a special set that has no elements. This set is called the empty set, or null set, and is denoted by ∅. q The empty set can also be denoted by { }. Ø A set with one element is called a singleton set.

Sets Equivalence A= {2, 3, 4, 5} B= { 2, 2, 3, 4, 4, 5}. Example : These two sets are equal because they have the same elements. A= {1, 2, 3} and B={x| x is a positive integer and x 2 <12} Then A= B

The size of a set q Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|. ü Remark: The term cardinality comes from the common usage of the term cardinal number as the size of a finite set. Example (1): Let A be the set of odd positive integers less than 10. Then |A|= 5. Example (2): Let S be the set of letters in the English alphabet. Then |S| = 26. Example (3): Because the null set has no elements, it follows that |∅| = 0

Some conclusions Ø For any sets A, B and C: q ∅ ⊂A ⊂ U q A⊆A q If A ⊂ B and B ⊂ C, then A ⊂ C q A = B if and only if A ⊆ B and B ⊆ A.

Set Operations q Union: A ⋃ B= { X: X∈ A or X ∈ B} Example : A= {a, b, c} and B= {b, c, d, e} Then A ⋃ B= { a, b, c, d, e}. q Intersection: A ⋂ B= { X: X∈ A and X ∈ B} Example : A= {a, b, c} and B= {b, c, d, e} Then A ⋂ B= {b, c}.

q Difference: A - B= { X: X∈ A and X ∉ B} Example : A= {1, 2, 3, 4, 5, 6} and B= {3, 5} Then A - B= {1, 2, 4, 6}. q Complement: AC= { X: X∈ U and X ∉ A} Example (1): U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A= {1, 3, 5, 7, 9} Then AC = {2, 4, 6, 8, 10}. Example (2): A= {1, 2, 3, 4, 5} and B= {2, 3, 4} Then BC = { 1, 5}.

Example Let A= {8, 7, 2}, E= { 1, 2, 3, 4, 5, 6, 7, 8, 9} and S= {2, 3, 5, 7}. Find AC, SC , EC , (A ⋂ S )C and (A ⋃ S )C ? AC= { 1, 3, 4, 5, 6, 9} SC = { 1, 4, 6, 8, 9} EC= { }. (A ⋂ S )C= {1, 3, 4, 5, 6, 8, 9} (A ⋃ S )C= { 1, 4, 6, 9}

Power Set q Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S). q If a set has n elements, then its power set has 2 n elements. Example (1) : What is the power set of the set S {0, 1, 2}? The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}. P (S) = 8. Example (2) : What is the power set of the empty set? What is the power set of the set {∅}? ü The empty set has exactly one subset, namely, itself. Consequently, P(∅) = {∅}. ü The set {∅} has exactly two subsets, namely, ∅ and the set {∅} itself. Therefore, P({∅}) = {∅, {∅}}.