l2_15a_2

# l2_15a_2 - ECE 15A Fundamentals of Logic Design Lecture 2...

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 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
3 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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 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
5 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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/29/2009 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

### Page1 / 18

l2_15a_2 - ECE 15A Fundamentals of Logic Design Lecture 2...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online