An algebraic structure An algebraic structure consists of
An algebraic structure • An algebraic structure consists of – a set of elements B – binary operators {+, . } – and a unary operator { ‘ } • Such that following holds – Membership: B contains at least two elements a and b – Closure: a+b is in B and a. b is in B – Commutativity: a+b = b+a and a. b = b. a – Associativity: a+(b+c)=(a+b)+c and a. (b. c) = (a. b). c – Identity a+0 = a and a. 1=a – Distributivity: a+(b. c) = (a+b). (a+c) and a. (b+c)=(a. b)+(a. c) – Complementarity: a+a’=1 and a. a’=0 1
Boolean algebra and its axioms and theorems • Besides being an algebraic structure other useful axioms and theorems are – Null: x+1 = 1 and x. 0 = 0 – Idempotency: x+x = x and x. x = x – Involution: (x’)’ = x – Uniting: x. y+x. y’=x and (x+y). (x+y’)=x – Absorption: x+x. y=x and x. (x+y)=x – (another form): (x+y’). y=x. y and (x. y’)+y=x+y – de Morgan’s: (x+y+. . )’=x’. y’. . . and (x. y…)’=x’+y’+…. – Generalized de Morgan’s: • f’(x 1, x 2, …, xn, 0, 1, +, . ) = f(x 1’, x 2’, …, xn’, 1, 0, . , +) 2
Duality axioms and theorems • Duality – A dual of a Boolean expression is derived by replacing. by +, + by. , 0 by 1, and 1 by 0 and leaving variables unchanged – Any theorem that can be proven is thus also proven for – a meta-theorem (a theorem about theorems) – duality: (x+y+…)D=x. y…… and (x. y…. )D = x+y+… – general duality: f. D(x 1, x 2, …, xn, 0, 1, +, . ) = f(x 1, x 2, …, xn, 1, 0, . , +) – multiplication and factoring: • (x+y). (x’+z) = x. z+x’. y and x. y+x’. z=(x+z). (x’+y) – Consensus: • (x. y)+(y. z)+(x’. z)=x. y+x’z and (x+y). (y+z). (x’+z)=(x+y). (x’+z) 3
Proving theorems • Prove x. y+x. y’=x • distributivity: • complementarity: • identity: • Prove x+x. y = x • identity: • distributivity: • identity: • NOR is equivalent of AND • (x+y)’ = x’. y’ • NAND is equivalent of OR • (x. y)’ = x’+y’ x. y+x. y’ = x. (y+y’) = x. (1) = x x+x. y = x. 1+x. y = x. (1+y) = x. (1) = x 4
And now try some harder problem • Simplify Boolean expression for carry function in a 3 -bit adder • Cout = a’. b. cin + a. b’. cin + a. b. cin’ + a. b. cin – Each of the first, second, and third term can be combined with the last term – Use identity to make copies of the last term 3 times (x+x=x) – Cout = a’. b. cin + a. b’. cin + a. b. cin’ + a. b. cin – Use associativity to bring terms together – Cout = a’. b. cin + a. b’. cin + a. b. cin’ + a. b. cin – Then use distributivity to combine terms – Cout = (a’+a). b. cin + (b’+b). a. cin + a. b. (cin’+cin) – Next use complementarity to reduce – Cout = (1). b. cin + (1). a. cin + a. b. (1) – Finally using identity gives Cout = b. cin + a. b 5
Multiple forms and equivalence • • Canonical Sum-of-Product form Canonical Product-of-sum form How to convert one from other? Minterm expansion of F to minterm expansion of F’ – Just take the terms that are missing • Maxterm expansion of F to maxterm expansion of F’ – Just take the terms that are missing 6
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