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- rules of evaluating a Boolean expression.

- Boolean Theorems.

- DeMorgan's Theorem.

- Universality of NAND and NOR Gates.

- Alternate Logic Gate Representations.

- Minterms and Maxterms.

- STANDARD FORMS.

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Proof:- x+xy = x.1+x.y = x(1+y) = x.1 = x

- 1. CHAPTER 3 Boolean Algebra
- 2. Contents Describing Logic Circuits Algebraically rules of evaluating a Boolean expression Boolean Theorems DeMorgan's Theorem Universality of NAND and NOR Gates Alternate Logic Gate Representations Minterms and Maxterms STANDARD FORMS 2
- 3. Describing Logic Circuits Algebraically OR gate, AND gate, and NOT circuit are the basic building blocks of digital systems 3
- 4. Circuits containing Inverters 4
- 5. Evaluating Logic Circuit Outputs Let A=0, B=0, C=1, D=1, E=1 X = [D+ ((A+B)C)'] • E = [1 + ((0+0)1 )'] • 1 = [1 + (0•1)'] • 1 = [1+ 0'] •1 = [1+ 1 ] • 1 =1 5
- 6. Rules of evaluating a Boolean expression First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0. Then perform all operations within parentheses. Perform an AND operation before an OR operation unless parentheses indicate otherwise. If an expression has a bar over it, perform the operations of the expression first and then invert the result. 6
- 7. Determining Output Level from a Diagram 7
- 8. Boolean Theorems 8
- 9. Multivariable Theorems (9) (10) (11) (12) (13.a) (13.b) (13.c) (14) (15) (16) x + y = y + x (Commutative law) x • y = y • x (Commutative law) x+ (y+z) = (x+y) +z = x+y+z (Associative law) x (yz) = (xy) z = xyz (Associative law) x (y+z) = xy + xz (Distributive law) x + yz = (x + y) (x + z) (Distributive law) (w+x)(y+z) = wy + xy + wz + xz x + xy = x (Absorption) [proof] x + x'y = x + y (x +y)(x + z) = x +yz 9
- 10. Proof of (14, 15, 16) x + xy x + x’y = x (1+y) = x • 1 [using theorem (6)] = x [using theorem (2)] = ( x + x’) (x + y) [theorem 13b] = 1 (x +y) = (x + y) (x +y)(x + z) =xx + xz + yx + yz = x + xz + yx + yz = x (1+z+y) +yz = x . 1 + yz = x + yz 10
- 11. DeMorgan's Theorem (18) (x+y)' = x' • y' (19) (x•y)' = x' + y' Example X = [(A'+C) • (B+D')]' = (A'+C)' + (B+D')' = (AC') + (B'D) = AC' + B'D 11
- 12. Three Variables DeMorgan's Theorem (20) (x+y+z)' = x' • y' • z' (21) (xyz)' = x' + y' + z‘ EXAMPLE: Apply DeMorgan’s theorems to each of the following expressions: (a) ( A + B + C) D (b) ABC + DEF (c) A B + CD + EF 12
- 13. Universality of NAND Gates 13
- 14. Universality of NOR Gates 14
- 15. Alternate Logic Gate Representations 15
- 16. Minterms and Maxterms x y z 0 0 0 Minterms Term Designation x' y’ z' m0 Maxterms Term Designation x+y+z M0 0 0 1 x' y' z M1 0 1 0 x' y z’ m2 0 1 1 x' y z m3 1 0 0 x y' z’ m4 1 0 1 x y' z m5 1 1 0 x y z’ m6 1 1 1 xyz m7 x+y+z’ +y x+y’+z +y -t- Z x+y’+z’ + y' +2 x'+y+z +y' + '+y+z’ xt , '+y + '+y’+z x z 4-'+y’+z’ x 2' M1 M2 M3 M4 M5 M6 M7 2 16
- 17. Canonical FORMS There are two types of canonical forms: the sum of minterms The product of maxterms 17
- 18. Sum of minterms f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7 f2 = x'yz + xy'z + xyz’ + xyz = m3 + m5+ m6 + m7 x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 f1 0 1 0 0 1 0 0 1 f2 0 0 0 1 0 1 1 1 18
- 19. Product of maxterms The complement of f1 is read by forming a minterm for each combination that produces a 0 as: f1’=x’y’z’ + x’yz’ + x’yz + xy’z + xyz’ f1 = (x + y + z)(x + y' + z)(x + y' + z' )(x’+ y + z)(x’ + y' + z) = Mo M2 M3 M5 M6 Similarly f2 = ? 19
- 20. Example: Sum of Minterms Express the Boolean function F = A + B'C in a sum of minterms. F=A+B'C = ABC + ABC' + AB'C + AB'C' + AB'C + A'B'C 20
- 21. Example: Product of Maxterms Express the Boolean function F =xy' + yz in a product of maxterm form. F = xy' + yz = (xy' + y)(xy' + z) = (x + y)(y' + y)(x + z)(y' + z) = (x + y)(x + z)(y' + z) = (x + y + zz')(x + yy' + z)(xx' + y' + z) = (x + y + z)(x + y + z')(x+y + z)(x+y’+ z)(x + y' + z)(x'+y'+z) = (x + y + z)(x + y + z') (x + y' + z) (x'+y'+z) = M0 M1 M2 M6 = Π (0,1,2,6) 21
- 22. STANDARD FORMS There are two types of standard forms: the sum of products (SOP) The product of sums (POS). 22
- 23. Sum of Products The sum of products is a Boolean expression containing AND terms, called product terms, of one or more literals each. The sum denotes the ORing of these terms. F = xy + z +xy'z'. (SOP) 23
- 24. Product of Sums A product of sums is a Boolean expression containing OR terms, called sum terms. Each term may have any number of literals. The product denotes the ANDing of these terms. F = z(x+y)(x+y+z) (POS) F = x (xy' + zy) (nonstandard form) 24

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