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Boolean Algebra•Purpose of BA is to facilitates design and analysis of digital circuits.
•For example the value of boolean function F=A + BC’ with the
following gate implementation can be shown by this truth table:
ABC
A+BC
A B C BC A+BC00001111
00110011
01010101
00100010
00101111
Boolean Algebra• Basic identities of boolean algebra
1. X 0 X
3. X 1 1
5. X X X 7. X X ’ 1excluded
middle
9. (X ’)’ X involution
10. X Y Y X
12. XYZ ) (XY )Z
14. XYZ ) XY XZ
16. X Y ) X Y 18. X + XY = X
2. X1 X identity
4. X0 0 base
6. XX X idempotence
8. XX’ 0 non contradiction
11. XY YX commutative
13. XYZ ) (XY )Z associative
15. XYZ ) XY )XZ ) distributive
17. XY)’ = XY demorgan
19. X.(X+Y) = x absorbtion
Basic identities of B.A. can be proven by truth table:
XYZ ) XY )XZ )YZ
00010001
XYZ000001010011100101110111
XYZ )00011111
XY00111111
XZ01011111
XY )XZ )00011111
X Y ) X Y XY00011011
X Y 1000
X Y )1000
demorgan
distributive
Boolean Algebra
• For each algebraic expression the dual of of algebraic expression achieved by interchanging AND and OR operators and replacing 0’s and 1’s.
• Parallel columns illustrate duality principle. The duality principle states that if E1 and E2 are Boolean expressions then
E1= E2 dual (E1)=dual (E2)where dual(E) is the dual of E
• Note: 15-17 have no counterpart in ordinary algebra.
• Other handy identity.X+X’Y=X+Y (15, 7 and 2)
Boolean Algebra
• By using boolean algebra rules, a simpler expression may be obtained
• Operator Precedence: when evaluating boolean expression , order of precedence is:
1. Parentheses
2. NOT
3. AND
4. OR
For example :Look at DeMorgan truth table first (x+Y) is computed then complement of (X+Y).
But for x’.y’ first the complement of x and complement of y is computed and then the result is ANDed
Boolean function simplification
• It means by manipulation of B.A. reducing the number of terms and literals in the function. For example:
f= x’y’z + x’yz + xy’
= x’(y’z + yz) + xy’
= x’ (z(y’+y)) + xy’
= x’(z.1) +xy’
= x’z + xy’
The Consensus Theorem
Theorem. XY +YZ + X Z = XY + X ZProof. XY +YZ + X Z = XY + (X + X )YZ + X Z
2,7
= XY + XYZ + X YZ + X Z 14
= XY(1 + Z ) + X Z(Y + 1) 2,11,14
= XY + X Z 3,2
Dual. X + Y )Y + Z )X + Z ) = X + Y )X + Z )
Complement of a Function
There are two ways for doing that:1) Using DeMorgan’s theorem2) Taking the dual of the function and
complement each literalFor example complements of x’yz’ + x’y’z = (x+y’+ z)(x+ y+z’) x(y’z’ + yz) = x’ + (y+z)(y’ + z’)
Canonical and standard Forms• The sum of products is one of two standard forms for Boolean
expressions.sum-of-products-expression = p-term + p-term ... + p-termp-term = literal literal literal– example. X Y Z + X Z + XY + XYZ
• A minterm is a term that contains every variable, in either complemented or un-complemented form.– example. in expression above, X Y Z is minterm, but X Z is
not
• A sum of minterms expression is a sum of products expression in which every term is a minterm.– example: X Y Z + X YZ + XYZ + XYZ is sum of minterms
expression that is equivalent to expression above.– shorthand : list minterms numerically, so X Y Z + X YZ +
XYZ + XYZ becomes 001+011+110+111 or m (1,3,6,7)
Canonical and standard Forms• The product of sums is the second standard form for Boolean
expressions.product-of-sums-expression = s-term s-term ... s-terms-term = literal literal literal– example. (X +Y +Z )(X +Z )(X +Y )(X +Y +Z )
• A maxterm is a sum term that contains every variable, in complemented or uncomplemented form.– example. in exp. above, X +Y +Z is a maxterm, but X +Z is
not
• A product of maxterms expression is a product of sums expression in which every term is a maxterm.– example. (X +Y +Z )(X +Y+Z )(X+Y+Z )(X+Y+Z ) is product of
maxterms expression that is equivalent to expression above.– shorthand : list maxterms numerically: so, (X +Y +Z )(X
+Y+Z) (X+Y+Z )(X+Y+Z ) becomes 110+100+001+000 or– M(6,4,1,0)
How to build the boolean function from truth table
• One way is to find a minterms or standard products by ANDing the terms of the n variable, each being primed if it is 0 and unprimed if it is 1. A boolean function can be formed by forming a minterm for each combination of variables that produce 1 in the function and then taking OR of all those forms.
• Another way is by finding maxterms or standard sums by OR term of the n variables, with each variable being unprimed if corresponding bit is 0 and primed if it is 1. A boolean function can be formed as a product of maxterms for each combination of variables that produce 0 in the function and then form And of all those forms
For example:
x y z function f1 function f2 minterms maxterms 0 0 0 0 0 m0 M0 0 0 1 1 0 m1 M1 0 1 0 0 0 m2 M2 0 1 1 0 1 m3 M3 1 0 0 1 0 m4 M4 1 0 1 0 1 m5 M5 1 1 0 0 1 m6 M6 1 1 1 1 1 m7 M7 Sum of mintermsf1 = x’y’z + xy’z’ + xyz = m1 + m4 + m7f2= x’yz + xy’z + xyz’ + xyx = m3 + m5 + m6 + m7
Product of maxtermsf1= (x+y+ z)(x+y’+z)(x’+ y + z’)(x’ + y + z) = M0M2M3M5M6f2= (x+y+z)(x+y+z’)(x+y’+z)(x’+y+z) = M0M1M2M4
Conversion between Canonical Forms
• By reading from a truth table the two canonical forms ( sum of minterms and product of maxterms) can be obtained.
• A boolean function can be converted to the canonical form. For example:
F= xy + x’y (in form of sum of the products S.O.P)
= (z+ z’)xy + x’z(y+y) = xyz + xyz’ + x’zy + x’zy’ can be converted to sum of minterms that can be shown by m1+m3+m6+m7 or
∑(1,3,6,7)
Conversion between Canonical Forms
• To convert it to the product of maxterms
F= xy + x’y = (xy + x’)(xy +z) =
(x’ + x) (x’+ y)(x+z)(y+z) = (x’+y)(x+z)(y+z)
it is in the form of products of sums (P.O.S) but we want the product of maxterms. So
= (x’+y+(z.z’)) (x + z + (y.y’))(y +z + (x.x’))
= (x’ + y + z)(x + z + y)(x+ z + y’)(x’ + y +z’)
= M0M2M4M5=∏(0,2,4,5)
Conversion between Canonical Forms• To convert from one canonical form to another,
interchange the symbol ∑ and ∏ and list those numbers missing from the total number of minterms or maxterms which is 2n,where n is number of variables.
• To verify that we convert sum of minterms to the product of maxterms by finding the complement of a function presented as sum of minterms. From the result the product of maxterms can be easily obtained
For example: F(x,y,z) = m1+m3+m6+m7= ∑(1,3,6,7) the complement of F (presented in the form of sum of the minterms) are the minterms that makes F to be zero thus
F’(x,y,z) = (∑(1,3,6,7))’= (m0 + m2 + m4 + m5) F = (F’(x,y,z))’ = (m0 + m2 + m4 + m5)’ Using Demorgan’s =m0’m2’m4’m5’= since each m’j = Mj it is M0M2M4M5 = ∏(0,2,4,5)
Standard forms• Sometimes boolean functions are shown as standard forms.
For example: F1 =y’ + xy + x’y’z’ ( sum of products) F2 = x(y’ + z) (x’ + y + z’) (product of sums) the product and sum can be used to make the gate structure
consist of AND and OR gates
• Sometimes boolean function can be shown in non standard forms:
F3= AB + C(D + E) can be changed to AB + CD + CE
• Different forms results different level of implementation of logical gates (see the next slide)