K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions.

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1 K-Map Lesso Objectves: Eve though Boolea expressos ca be smplfed by algebrac mapulato, such a approach lacks clear regular rules for each succeedg step ad t s dffcult to determe whether the smplest expresso has bee acheved. I cotrast, Karaugh map (K-map) method provdes a straghtforward procedure for smplfyg Boolea fuctos. K-maps of up to 4 varables are very commo to use. Maps of 5 ad 6 varables ca be made as well, but are more cumbersome to use. Smplfed expressos produced by K-maps are always ether the SOP or the POS form. The map provdes the same formato cotaed a Truth Table but a dfferet format. The objectves of ths lesso are to lear:. How to buld a 2, 3, or 4 varables K-map. 2. How to obta a mmzed SOP fucto usg K-maps. Code Dstace: Let s frst defe the cocept of Code Dstace. The dstace betwee two bary code-words s the umber of bt postos whch the two code-words have dfferet values. For example, the dstace betwee the code words 00 ad 000 s whle the dstace betwee the code-words 00 ad 000 s 3. Ths defto of code dstace s commoly kow as the Hammg dstace betwee two codes. Two-Varable K-Maps: The 2-varable map s a table of 2 rows by 2 colums. The 2 rows represet the two values of the frst put varable A, whle the two colums represet the two values of the secod put varable B. Thus, all etres (squares) the frst row correspod to put varable A=0, whle etres (squares) of the secod row correspod to A=. Lkewse, all etres of the frst colum correspod to put varable B = 0, whle etres of the secod colum correspod to B=. Thus, each map etry (or square) correspods to a uque value for the put varables A ad B.

2 For example, the top left square correspods to put combato AB=00. I other words, ths square represets mterm m 0. Lkewse, the top rght square correspods to put combato AB=0, or mterm m ad the bottom left square correspods to put combato AB=0, or mterm m 2. Fally, the bottom rght square correspods to put combato AB=, or mterm m 3. I geeral, each map etry (or square) correspods to a partcular put combato (or mterm). Sce, Boolea fuctos of two-varables have four mterms, a 2-varable K-map ca represet ay 2-varable fucto by pluggg the fucto value for each put combato the correspodg square. Deftos/Notatos: Two K-map squares are cosdered adjacet f the put codes they represet have a Hammg dstace of. A K-map square wth a fucto value of wll be referred to as a -Square. A K-map square wth a fucto value of 0 wll be referred to as a 0-Square. The smplfcato procedure s summarzed below: Step : Draw the map accordg to the umber of put varables of the fucto. Step 2: Fll s the squares for whch the fucto s true. Step 3: Form as bg group of adjacet -squares as possble. There are some rules for ths whch you wll lear wth bgger maps. Step 4: Fd the commo lterals for each group ad wrte the smplfed expresso SOP. Example: Cosder the gve truth table of two varable fucto. Obta the smplfed fucto usg K-map. A B F

3 Frst draw a 2-varable K-map. The fucto F s true whe AB (m 2 ) s true ad whe AB (m 3 ) s true, so a s placed sde the square that belogs to m 2 ad a s placed sde the square that belogs to m 3. Sce both of the -squares have dfferet values for varable B but the same value for varable A, whch s,.e., wherever A = the F = thus F = A. Ths smplfcato s justfed by algebrac mapulato as follows: F = m 2 + m 3 = AB + AB = A (B + B) = A To uderstad how combg squares smplfes Boolea fuctos, the basc property possessed by the adjacet squares must be recogzed. I the above example, the two -squares are adjacet wth the same value for varable A (A=) but dfferet values for varable B (oe square has B=0, whle the other has B=). Ths reducto s possble sce both squares are adjacet ad the et expresso s that of the commo varable (A). Geerally, ths s true for ay 2 codes of Hammg dstace (adjacet). For a - varable K-map, let the codes of two adjacet squares (dstace of ) have the same value for all varables except the th varable. Thus, Code of st Square: Code of 2 d Square: Combg these two squares a group wll elmate the dfferet varable ad the combed expresso wll be..... sce: ( ) ( ) (..... ) ( ) = ( + ) =. 2 The varable dfferece s dropped.. 2 Aother Example: Smplfy the gve fucto usg K-map method: F = (, 2, 3)

4 I ths example: F = m + m 2 + m 3 = m + m 2 + (m 3 + m 3 ) F = (m + m 3 ) + (m 2 + m 3 ) = A + B Rule: A -square ca be member of more tha oe group. If we exchage the places of A ad B, the mterm postos wll also chage. Thus, m ad m 2 wll be exchaged as well. I a -varable map each square s adjacet to other squares, e.g., a 2-varable map each square s adjacet to two other squares as show below: Examples of o-adjacet squares are show below:

5 Three-Varable K-Maps: There are eght mterms for a Boolea fucto wth three-varables. Hece, a threevarable map cossts of 8 squares. All etres (squares) the frst row correspod to put varable A=0, whle etres (squares) of the secod row correspod to A=. Lkewse, all etres of the frst colum correspod to put varable B = 0, C = 0, all etres of the secod colum correspod to put varable B = 0, C =, all etres of the thrd colum correspod to put varable B =, C =, whle etres of the fourth colum correspod to B=, C = 0. To mata adjacet colums physcally adjacet o the map, the colum coordates do ot follow the bary cout sequece. Ths choce yelds ut dstace betwee codes of oe colum to the ext (00 0 0), lke Grey Code. Varatos of Three-Varable Map: The fgure shows varatos of the three-varable map. Note that the mterm correspodg to each square ca be obtaed by substtutg the values of varables ABC order. Examples: (see authorware verso)

6 There are cases where two squares the map are cosdered to be adjacet eve though they do ot physcally touch each other. I the fgure of 3-varable map, m 0 s adjacet to m 2 ad m 4 s adjacet to m 6 because the mterms dffer by oly oe varable. Ths ca be verfed algebracally: m 0 + m 2 = A B C + A BC = A C (B + B) = A C m 4 + m 6 = AB C + ABC = AC (B + B) = AC Rule: Groups may oly cosst of 2, 4, 8, 6, squares (always power of 2). For example, groups may ot cosst of 3, 6 or 2 squares. Rule: Members of a group must have a closed loop adjacecy,.e., L-Shaped 4 squares do ot form a vald group. Notes:. Each square s adjacet to 3 other squares. 2. Oe square s represeted by a mterm (.e. a product term cotag all 3 lterals). 3. A group of 2 adjacet squares s represeted by a product term cotag oly 2 lterals,.e., lteral s dropped. 4. A group of 4 adjacet squares s represeted by a product term cotag oly lteral,.e., 2 lterals are dropped. Four-Varable K-Maps: There are 6 mterms for a Boolea fucto wth four-varables. Hece, fourvarable map cossts of 6 squares.

7 Notes:. Each square s adjacet to 4 other squares. 2. Oe square s represeted by a mterm (a product of all 4-lterals). 3. Combg 2 squares drops -lteral. 4. Combg 4 squares drops 2-lterals. 5. Combg 8 squares drops 3-lterals. Examples: (see authorware verso) Rule: The combato of squares that ca be chose durg the smplfcato process the -varable map are as follows: A group of 2 squares produces a fucto that always equal to logc. A group of 2 - squares represets a product term of oe lteral. A group of 2-2 squares represets a product term of two lterals ad so o. Oe square represets a mterm of lterals.