DE_MF

4.3 Rules and Laws of Algebra 1. CommutA+B=B+A (CommuAB=BA (Commutat

Booleantative Lawsutative law of addition)tive law of multiplication) 15

2. AssociaA+ (B+C) = (A+B) +C ( A (BC) = (AB) C (Assoc

ative laws(Associative law of addition)ciative law of multiplication) 16

3. Distribu A (B+C) =

utive Law= AB+AC 17

4.4 De Morgan's The Sometimes it is more econ the complement of a func result) than it is to implemen DeMorgan’s law provides complement of a Boolean fu Recall DeMorgan’s law stat

eoremnomical to build a circuit usingction (and complementing its nt the function directly. an easy way of finding the unction.tes: 18

 DeMorgan’s law can be e variables. Replace each variable by its ANDs to ORs and all ORs to Thus, we find the complemen is:

extended to any number ofs complement and change all ANDs.nt of: 19

 Through our exercises in si we see that there are nume Boolean expression. – These “synonymous” forms are – Logically equivalent expression In order to eliminate as designers express Boolean canonical form.

implifying Boolean expressions,erous ways of stating the samee logically equivalent.ns have identical truth tables. much confusion as possible,n functions in standardized or 20

 There are two canonical forms products and product-of-sums. • Recall the Boolean product is sum is the OR operation. In the sum-of-products form, ANDe For example: In the product-of-sums form, ORed For example:

for Boolean expressions: sum-of-the AND operation and the Booleaned variables are ORed together.d variables are ANDed together: 21

 It is easy to convert a function to of-products form using its truth tab We are interested in the values variables that make the functio (=1). Using the truth table, we list the of the variables that result in function value. Each group of variables is then together.

o sum-ble. of theon true values a true ORed 22

• The sum-of-products fo for our function is: We note that this function not in simplest terms. O aim is only to rewrite o function in canonical sum-o products form.

orm isOurour of- 23

4.5 Karnaugh Map The Karnaugh map, also known as boolean algebra expressions. The Karnaugh map reduces the need advantage of humans' pattern-recogniti The required Boolean results are trans dimensional grid where the cells are position represents one combination of represents the corresponding output v identified. These terms can be used to write a mi the required logic.

p the K-map, is a method to simplify d for extensive calculations by taking ion capability. sferred from a truth table onto a two- ordered in Gray code, and each cell f input conditions, while each cell value value. Optimal groups of 1s or 0s are inimal boolean expression representing 24

Karnaugh Maps - Rule Simplification The Karnaugh map uses simplification of expressio adjacent cells containing on1.Groups may not include an

es ofs the following rules for theons by grouping together nes.ny cell containing a zero. 25

2.Groups may be horizontal diagonal.

l or vertical, but not 26

3. Groups must contain 1, 2, 4, 8, That is if n = 1, a group will con If n = 2, a group will contain fou

or in general 2n cells.ntain two 1's since 21 = 2.ur 1's since 22 = 4. 27

4. Each group should be as

s large as possible. 28

5.Each cell containing a one m

must be in at least one group. 29

6.Groups may overlap.

30