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Algebra

Algebra. Algebra – defined by the tuple:  A , o 1 , …, o k ; R 1 , …, R m ; c 1 , …, c k  Where: A is a non-empty set o i is the function, o i : A p i  A where p i is a positive integer R j is a relation on A c i is an element of A EXAMPLE

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Algebra

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  1. Algebra Algebra – defined by the tuple: A, o1, …, ok; R1, …, Rm; c1, …, ck Where:A is a non-empty setoi is the function, oi: Api A where piis a positive integerRj is a relation on Aci is an element of A EXAMPLE Z, +,  Zis a set of integers + is addition operation  is “less than or equal to” relation

  2. Lattice Algebra Lattice Algebra – defined by the tuple: A, , • Where:A is a non-empty set , • are binary operations And, the Following Axioms Hold: a  a = aa• a = a (Idempotence) a  b = b  aa•b = b•a (Commutativity) a  (b  c) = (a  b )  ca• (b• c) = (a•b) •c (Associativity) a  (a•b) = aa• (a  b) = a (Absorption) a,b,c  A

  3. Distributive Lattice Algebra Distributive Lattice Algebra A Lattice Algebra plus the Following Distributive Laws Hold: a  (b•c) = (a  b ) • (a  c) a• (b  c) = (a•b) (a•c) Complemented Distributive Lattice Algebra 1) maximal element = 1 2) minimal element = 0 3) For anya  Aif  xa Asuch thata• xa = 0 4) For anya  Aif  xa Asuch thata  xa = 1 A Complemented Distributive Algebra is a Boolean Algebra

  4. Distributive Lattice Examples 1 c is complement of a c is complement of b 1 a c No complement b a 0 0 a• (b  c) = (a•b) (a•c)? a• 1 = a (a•b) (a•c) = b 0 = b No, non-distributive lattice!

  5. Boolean Algebra B, , •, , 0, 1 0, 1 B is a unary operation over B , • are binary operations overB 0 is the “identity element”wrt 1 is the “identity element”wrt • Ordered Set Lattice Dist. Lattice Boolean Algebra

  6. Boolean Algebra Postulates B, , •, , 0, 1 0, 1 B For arbitrary elements a,b,c  B the Following Postulates Hold: Absorption a  (a•b) = aa• (a  b) = a Associativitya  (b  c) = (a  b )  ca• (b•c ) = (a•b) •c Commutativitya  b = b  aa•b = b•a Idempotencea  a = aa•a = a Distributivitya  (b•c) = (a  b) • (a  c) a• (b  c) = (a•b) (a•c) Involutiona = a Complementa  a = 1 a•a = 0 Identitya  0 = aa• 1 = aa  1 = 1 a• 0 = 0 DeMorgan’sa  b = a• ba•b = a  b

  7. Huntington’s Postulates B, , •, , 0, 1 0, 1 B All Previous Postulates may be Derived Using: Commutativitya  b = b  aa•b = b•a Distributivitya  (b•c) = (a  b) • (a  c) a• (b  c) = (a•b) (a•c) Complementa  a = 1 a•a = 0 Identitya  0 = aa• 1 = a If Huntington’s Postulates Hold for an Algebra then it is a Boolean Algebra

  8. DeMorgan’s Theorem B, , •, , 0, 1 0, 1 B Theorem: Let F(x1, x2,…,xn) be a Boolean expression. Then, the complement of the Boolean expression F(x1, x2,…, xn) is obtained from F as follows: 1) Add parentheses according to order of operation 2) Interchange all occurrences of  with • 3) Interchange all occurrences of xi with xi 4) Interchange all occurrences of 0 with 1 EXAMPLEF =a  ( b • c ) F = a• ( b  c ) a  (b • c ) = a• ( b  c )

  9. Principle of Duality B, , •, , 0, 1 0, 1 B Interchanging all occurrences of  with • and/or interchanging all occurrences of 0 with 1 in an identity, results in another identity that holds. A is a Boolean expression and AD is the Dual of A 0D=1 1D=0 A, Band Care Boolean Expressions ifA = B  CthenAD = BD• CD if A = B•CthenAD = BD CD ifA = B then AD = BD

  10. Logic (Switching) Functions B ={0, 1} The set of all mappings Bn B for B ={0,1} can be represented by Boolean expressions and are called “two-valued logic functions” or “switching functions”. The set Bn contains 2n elements The total number of mappings or functions is The notation we use is f: Bn B f can also be described through the use of an expression

  11. Multi-dimensional Logic Functions • f:BnBmB={0,1} • fis a vector of functionsfi: BnBwhere I = 1 to m • Bnrepresents the set of all elements in the set formed bynapplications of the Cartesian ProductB B  …  B • Bncan also be interpreted geometrically as ann-dimensionalhypercube • The geometrical representations are referred to as “cubical representations” • Each element inBnis represents a geometric coordinate a discrete hyperspace

  12. Cubical Representation • Considerf:B3BB={0,1} • The domain offis a hypercube of dimensionn = 3 • The range of f is a hypercube of dimension n = 1 (0,1,1) (1,1,1) (1,1,0) f (0,1,0) 0 1 (0,0,1) (1,0,1) (0,0,0) (1,0,0) NOTE: These are (sideways) Hasse Diagrams for B3 and B1 !!!

  13. Some Definitions • variable – A symbol representing an element of B • xi B • literal – xi or xi • if xi=0 then xi=1 • if xi=1 then xi=0 • product – a Boolean expression composed of literals and the  operator • (e.g.x1x3x4) • NOTE: when two literals appear next to each other, the  operation is “assumed” to be present • (e.g.x1x3x4) • cube – another term for a product • minterm – an element ofBnfor f:Bn Bsuch thatf = 1 • j-cube – a product composed ofn-jliterals • f(x1, x2,…,xn) – a functionf: Bn  B • f(x1, x2,…, xn) – a multi-dimensional functionf : Bn Bm

  14. Functions and Expressions B, +, •, , 0, 1 B ={0, 1} A specific function may be defined by an expression EXAMPLE: Consider the Boolean algebra defined above. Each operation can be given a name and defined by a personality matrix or table. The table contains the operation result for each element in BB for a binary operation and for each element in B for a unary operation. NAME is OR NAME is AND NAME is NOT EXAMPLE: A function f: B3 B can be specified by an expression over some Boolean algebra. f(x1, x2, x3) = x1 + x1x2x3

  15. Geometric Interpretation of a Function B, +, •, , 0, 1 B ={0, 1} f(x1, x2, x3) = x1 + x1x2 x3 (0,1,1) (1,1,1) (1,1,0) f (0,1,0) x2 0 1 x3 (0,0,1) (1,0,1) x1 (0,0,0) (1,0,0) f 1denotes the “on-set” of the functionf “on-set” is a set of cubes in the domain offfor whichf = 1 f 1={x1, x1x2x3}

  16. Geometric Description of a Function  B, +, •, , 0, 1 B ={0, 1} f(x1, x2, x3) = x1 + x1x2x3 (0,1,1) (1,1,1) (1,1,0) (0,1,0) x2 x3 (0,0,1) (1,0,1) x1 (0,0,0) (1,0,0) • f = (0,1,2,3,6) where each value represents a 0-cube • Each cube inf1={x1, x1x2x3} “covers” 1 or more 0-cubes • x1 covers {000, 001, 010, 011} x1is a 2-cube • x1x2x3covers {110} x1x2x3is a 0-cube

  17. Geometric Description (cont)  B, +, •, , 0, 1 B ={0, 1} f(x1, x2, x3) = x1 + x1x2x3 =x1 + x2x3 (0,1,1) (0,1,1) (1,1,1) (1,1,1) (1,1,0) (1,1,0) (0,1,0) (0,1,0) x2 (0,0,1) x3 (0,0,1) (1,0,1) (1,0,1) x1 (0,0,0) (1,0,0) (0,0,0) (1,0,0) • minterm – any 0-cube that is covered by any element inf 1 • “don’t care” is a variable that is not present in a cube in f 1 • don’t care is denoted by “-” • f 1={x1--, x1x2x3}={-x2x3, x1-x3, x1x2x3} • x2 andx3 are don’t cares in cubex1

  18. Cube Sets • f 1is a set of all cubes for whichf = 1 (on-set) • f 0is a set of all cubes for whichf = 0 (off-set) • f DCis a set of all cubes for whichf = don’t care (DC-set) • fis “completely specified” if any two off 0, f 1orf DCare given • fis “incompletely specified” proper subsets are given forf 0,f 1 or f DC • f 1is said to be a “cover” forf

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