Truth Tables (again) Truth Table to SOP Form. Truth Table to POS Form. Recall that a boolean equation can be represented by a Truth Table

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1 Truth Tables (again) Recall that a boolean equation can be represented by a Truth Table C F truth table for a boolean function of N variables has 2 N entries. The s represent F(,,C). The 0 s represent F (,,C) R 2//99 Truth Table to SOP Form Can write SOP form of equation directly from truth table. C F C C C C C F(,,C) = C + C + C + C + C Note that each term in has LL variables present. If a product term has LL variables present, it is a MINTERM. R 2//99 2 Truth Table to POS Form To get POS form of F, write SOP form of F, then use DeMorgan s Law. F (,,C) = C + C + C Take complement of both sides: C F C (F (,,C)) = ( C + C + C ) C pply DeMorgan s Law to right side C Left side is (F ) = F F(,,C) = ( C ) ( C) ( C ) 0 apply DeMorgan s Law to each term 0 F(,,C) = (++C) (++C )(+ +C) POS Form!! R 2//99 3

2 Minterms, Maxterms We saw that: F(,,C) = C + C + C + C + C + C SOP form. If a product term has all variables present, it is a MINTERM. F(,,C) = (++C) (++C )(+ +C) POS form. If a sum term has all variables present, it is a MXTERM. ll oolean functions can be written in terms of either Minterms or Maxterms. R 2//99 4 Minterm, Maxterm Notation Each line in a truth table represents both a Minterm and a Maxterm. Row No. C Minterms Maxterms C = m 0 ++C = M C = m ++C = M C = m 2 + +C = M C = m 3 + +C = M C = m 4 ++C = M C = m 5 ++C = M C = m 6 + +C = M 6 7 C = m 7 + +C = M 7 R 2//99 5 Using Minterms, Maxterms boolean function can be written in terms of Minterm or Maxterm notation as a shorthand method of specifying the function. F(,,C) = C + C + C + C + C + C = m 3 + m 4 + m 5 + m 6 + m 7 = Σ m(3,4,5,6,7) F(,,C) = (++C) (++C )(+ +C) = M 0 M M 2 = M(0,,2) Minterms correspond to s of F, Maxterms correspond to 0 s of F in truth table. R 2//99 6 2

3 From Minterms to Truth Table Minterms correspond to s in Truth table F(,,C) = Σ m(,2,6) = m + m 2 + m 6 = C + C + C m m 2 m 6 C F R 2//99 7 From Minterms to Maxterms to Truthtable To go from Minterms to Maxterms, list the numbers that are NOT present (with 3 variables, minterm/maxterm numbers range from 0 to 7 F(,,C) = Σ m(,2,6) = M(0,3,4,5,7) = (++C)(+ +C )( ++C) ( ++C )( + +C ) M 0 M 3 M 4 M 5 M 7 Maxterms correspond to 0 s in Truth table C F R 2//99 8 Examples F(,,C,D) = Σ m(0) (minterm form) = C D (SOP form) = M(,2,3,4,5,6,7,8,9,0,,2,3,4,5) (maxterm form) (POS form too long to write..) F(,) = Σ m(,2) (minterm form) = + (SOP form) = M(0,3) (maxterm form) = (+)( + ) (POS form) = xor (did you recognize this?) R 2//99 9 3

4 Minterm Expansion minterm must have every variable present. If a boolean product term does not have every variable present, then it can be expanded to its minterm representation. F(,,C) = + C neither, or C are minterms To expand to minterms, use the relation: = (C+ C ) = C + C To expand C to minterms, do: C = C(+ ) = C+ C = C(+ ) + C(+ ) = C + C + C + C F = +C = C + C + C + C + C F(,,C) = Σ m(,3,5,6,7) R 2//99 0 Maxterm Expansion maxterm must have every variable present. If a boolean sum term does not have every variable present, then it can be expanded to its maxterm representation. F(,,C) = (+) (C) neither (+), or C are maxterms To expand (+) to maxterms, use the relation: (+) = (++C C) = (++C )(++C) To expand C to minterms, do: C = C+ = ( +C)(+C) = ( + +C)(+C+ ) = ( + +C)( ++C)(+ +C)(++C) F = (+)(C) = (++C)(++C )(+ +C)( ++C)( + +C) F(,,C) = Π M(0,,2,4,6) R 2//99 Minimize from Minterm From Y = Σ m(3,4,5,6,7) Y = C + C + C + C + C Look for differences in only one variable Y = C + (C + C) + (C + C) = C + + = C + ( +) = C + = C + difference in only one variable is called a oolean djacency. R 2//99 2 4

5 Minimize from POS Y = Π M(0,,2) Y = (++C)(++C )(+ +C) gain, look for differences in only one variable Y = (+ + CC ) (++C) = (+)(+ +C) = (+)((+C) + ) = (+)(+C) + (+) = + C + + = + + C = ( + ) + C = + C R 2//99 3 Karnaugh Maps Karnaugh Maps (K-Maps) are a graphical method of visualizing the 0 s and s of a boolean function K-Maps are very useful for performing oolean minimization. Will work on 2, 3, and 4 variable K-Maps in this class. Karnaugh maps can be easier to use than boolean equation minimization once you get used to it. R 2//99 4 K-Maps K-map has a square for each or 0 of a boolean function. One variable K-map has 2 = 2 squares. Two variable K-map has 2 2 =4 squares Three variable K-map has 2 3 = 8 squares Four variable K-map has 2 4 = 6 squares variable 3 variable 2 variable 4 variable R 2//99 5 5

6 Plotting Functions on K-Maps Each square represents a row in the truth table. The values in each square is the value of F from the truth table. Row F() 0 0?? =0 = =0 = r0 r Row F() Row F() =0 = 0 =0 = 0 F() = F() = R 2//99 6 Plotting 2-Variable Functions Row F(,) 0 0 0? 0? 2 0? 3? Row F(,) r0 r2 r F(,) = + r3 Row 0 from TT, =0, = 0 R 2//99 7 Plotting 2-Variable Functions (cont.) Row F(,) F(,) = + Row F(,) F(,) = R 2//99 8 6

7 Plotting 3-Variable Functions Row C F(,,C)? 0 0? 2 0 0? 3 0? 4 0 0? 5 0? 6 0? 7? C C 0 00 r0 r4 0 r r5 r3 r7 0 r2 r6 R 2//99 9 oolean djacency Note on the three variable map: C 0 00 r0 r4 0 r r5 r3 r7 0 r2 r6 C 0 00 r0 0 r 0 r2 r3 r4 r5 r6 r7 Correct WRONG!!! Each square on the 3-variable map is oolean djacent. djacent squares only differ by ONE OOLEN VRILE!!! R 2//99 20 oolean djacency C 0 00 f( C ) 0 f( C) f( C) 0 f( C ) f( C ) f( C) f(c) f(c ) Squares at bottom of map adjacent to squares top of map. Each square is boolean adjacent to neighbor. R 2//99 2 7

8 Plotting 3-Variable Functions Row C F(,,C) C F(,,C) = Σ m(0,2,6) R 2//99 22 Row C F(,,C) F(,,C) = Σ m(4,5,6) nother 3-variable Example C R 2//99 23 Row C D F(,,C,D) 0? 0 0 0? ? 3 0 0? ? 5 0 0? 6 0 0? 7 0? ? 9 0 0? 0 0 0? 0? 2 0 0? 3 0? 4 0? 5? Plotting 4-Variable Functions CD CD r0 r4 r2 r8 0 r r5 r3 r9 r3 r7 r5 r 0 r2 r6 0 r4 r0 R 2//

9 CD f( C D ) f( C D ) 0 f( C D) f( CD) f( C D) f( CD) 0 f( CD ) oolean djacency f( CD ) 0 f(c D ) f(c D) f(cd) f(cd ) f( C D ) f( C D) f( CD) f( CD ) Squares at bottom of map adjacent to squares top of map and viceversa. Squares at left edge are adjacent to squares at right edge and viceversa. R 2//99 25 Plotting 4-Variable Functions Row C D F(,,C,D) CD F =Σ m(2,3,6,0,5) R 2//99 26 What do you need to Know? Minterm, Maxterm definitions Truth table to Minterms, vice versa Truth table to Maxterms, vice versa Minterms to Maxterms, vice versa Plotting 2,3,4 variable functions on K-Maps R 2//