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l2_15a_6 - Today The Algebra of Sets The Algebra of Sets is...

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1 ECE 15A Fundamentals of Logic Design Lecture 2 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Today: The Algebra of Sets The Algebra of Sets is an example of Boolean Algebra useful in manipulating digital circuits We will investigate the nature of sets and the way in which they may be combined 3 Definitions Elements are basic objects Collections of objects constitute sets Basic objects Example: S 1 S 2 S 3 S 4 Sets S 5 Universal set, denoted by “1”: Consists of all elements under consideration Null set, denoted by “0”, Contains no elements 4 Sets Specification of a set: enumeration Example: {1,3,5,7} Order of set elements is arbitrary S1 = {pink, blue, red} S2 = {red, blue, pink} S1 = S2 A set is an unordered collection of elements The algebra will be developed as an algebra for sets, not for elements of sets 5 Empty set, Universal set A set which contains no elements is called the empty set, or the Null set. Often denoted as 0, { }. We will denote it as 0. For any element x 0. [x is not an element of 0] A set which is a collection of all elements under consideration forms the universal set We will denote this set as 1. 6 Subsets If sets A and B have the same elements: A=B; otherwise A = B. If all elements of A are elements of B, A is a subset of B: A B. B is a superset of A. If all elements of A are elements of B and there is an element in B which is not in A: A is a proper subset of B: A B.
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2 7 Examples Every set is a subset of 1; 0 is a subset of every other set. 0 and 1 are not numbers! Basic objects Example, continues: S 1 S 2 S 3 S 4 S 5 Universal set 1 m m is a member of S 4 S 2 S 1 m S 4 S 2 S 1 S 4 S 5 = p ={p} S 3 r k u = {r,k,u} S 2 S 6 S’ 6 (also denoted as S 6 ) Complement of a set S 6 8 Example: red, black and yellow books The red (R) books and some of the black books (B) are in English (E); The reminder of the black books (B) are in German (G); Yellow books (Y) are in French (F). Y R B E G F Y=F B G E R 9 Rules of forming new sets The union of sets X and Y is a set X+Y consisting of all elements which are either in X or in Y.
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