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Examples

Examples. Simplify: ab’c + abc + a’bc ab’c + abc + a’bc = ab’c + abc + abc + a’bc = ac + bc Show that X + X’Y = X + Y X + X’Y = X(1 + Y) + X’Y = X + XY + X’Y = X + Y. Examples (cont’d). Simplify:WX + XY + X’Z’ + WY’Z’ WX + XY + X’Z’ + WY’Z’ = WX + XY + X’Z’ + WY’Z’X + WY’Z’X’

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Examples

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  1. Examples • Simplify: ab’c + abc + a’bc ab’c + abc + a’bc = ab’c + abc + abc + a’bc = ac + bc • Show that X + X’Y = X + Y X + X’Y = X(1 + Y) + X’Y = X + XY + X’Y = X + Y

  2. Examples (cont’d) • Simplify:WX + XY + X’Z’ + WY’Z’ WX + XY + X’Z’ + WY’Z’ = WX + XY + X’Z’ + WY’Z’X + WY’Z’X’ = WX(1 + Y’Z’) + XY + X’Z’(1 + WY’) = WX + XY + X’Z’

  3. Examples (cont’d) • Prove the consensus theorem, which says: XY + X’Z + YZ = XY + X’Z • Solution: XY + X’Z + YZ = XY + X’Z + (X + X’)YZ = XY + X’Z + XYZ + X’YZ = XY + XYZ + X’Z + X’ZY = XY(1 + Z) + X’Z(1 + Y) = XY + X’Z

  4. Examples (cont’d) • Simplify: d(a+bc) + a’(b’ + c’)(d + e’) + e’(a + bc) d(a+bc) + a’(b’ + c’)(d + e’) + e’(a + bc) = (a + bc)(d + e’) + a’(b’ + c’)(d + e’) = (d + e’)(a + bc) + (a + bc)’(d + e’) = d + e’

  5. Implicants • An on-set member of any combinable group basically a product term e.g. A’BC,A,BC • Prime Implicant: cannot combine with other to eliminate a literal • Each corresponds to Prod term in min S-o-P • Essential Prime Implicant: If it is the only PI covering a particular minterm

  6. Example of Implicants • Implicants • Six Prime Implicants: A’B’D, BC’, AC, A’C’D, AB, B’CD • Essential PI: AC,BC’ • F=A’B’D+BC’+AC

  7. Karnaugh Maps • What was the idea in doing simplication? Well, one of the ideas was to try to apply the unification theorem (AB + AB’ = A). • What we’re looking for then are terms that differ only in one variable. • This can be difficult to do when there are many terms and many variables. Let’s try to see if we there is a graphical method that makes it easier.

  8. Cube Representation. • Let’s draw a cube. Each vertex is a possible term, AND two adjacent vertices only differ in one variable. • Now, draw a dot for each term from our boolean expression, and group dots that are connected. • An edge that connects two dots means that we can apply the unification theorem to merge those two terms. The variable that differs is dropped. • By applying the unification theorem twice, we can merge 4 vertices that are fully connected. • More generally, by applying the unification theorem 2n times, we can merge n vertices. A’BC ABC A’BC’ ABC’ B A’B’C C AB’C A AB’C’ A’B’C’ The above cube shows the expression A’BC + ABC’ + ABC + AB’C + AB’C’. It simplifies to: A + BC’

  9. Karnaugh Maps (cont’d) Cut here and unwind the cube to get the bottom diagram A’BC ABC A’BC’ ABC’ B A’B’C C AB’C A AB’C’ A’B’C’ A’BC A’B’C AB’C ABC Note how, because we are now in 2d, the lines kinda wrap around. A’BC’ A’B’C’ AB’C’ ABC’

  10. Karnaugh Maps (cont’d) A’BC A’B’C AB’C ABC A’BC’ A’B’C’ AB’C’ ABC’ A simpler drawing of this is given below. We call this kind of table a Karnaugh map.

  11. K-map : SoP and PoS • SoP: A’BC’D’+A’B’C’D+ABC’D+AB’C’D+A’B’CD+AB’CD • Minimized Exp A’BC’D’+B’D+AC’D • PoS: (A+B+C+D)(A’+B’+C+D)(A’+B+C+D) (A+B’+C+D’) …. • Minimized PoS (B+D)(A’+D)(B’+C’)(A+B’+D’)

  12. Detecting XOR on K-maps for 2 vars A xor B So, what we are looking for are diagonals

  13. Detecting XOR on K-maps for 3 vars A xor B: Have alternate columns of 1’s A xor C: Have diagonal groups of 1’s

  14. Detecting XOR on K-maps for 4 vars A xor B: Have alternate columns of 1’s A xor C: Have diagonal groups of 1’s

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