l2_15a_6

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

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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 ² an example of Boolean Algebra ² 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 ² Example: {1,3,5,7} ± Order of set elements is arbitrary ² S1 = {pink, blue, red} ² S2 = {red, blue, pink} ² 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. ² Often denoted as 0, { }. ² We will denote it as 0. ² 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 ² 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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This note was uploaded on 01/20/2011 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

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

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