An algebraic structure consists of

1
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 ab is in B and a.b is in B
• Commutativity ab ba and a.b b.a
• Associativity a(bc)(ab)c and a.(b.c)
  (a.b).c
• Identity a0 a and a.1a
• Distributivity a(b.c) (ab).(ac) and
  a.(bc)(a.b)(a.c)
• Complementarity aa1 and a.a0

2
Boolean algebra and its axioms and theorems

• Besides being an algebraic structure other useful
  axioms and theorems are
• Null x1 1 and x.0 0
• Idempotency xx x and x.x x
• Involution (x) x
• Uniting x.yx.yx and (xy).(xy)x
• Absorption xx.yx and x.(xy)x
• (another form) (xy).yx.y and (x.y)yxy
• de Morgans (xy..)x.y... and
  (x.y)xy.
• Generalized de Morgans
• f(x1,x2,,xn,0,1,,.) f(x1,x2,,xn,1,0,.,)

3
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 (xy)Dx.y and (x.y.)D xy
• general duality fD(x1,x2,,xn,0,1,,.)
  f(x1,x2,,xn,1,0,.,)
• multiplication and factoring
• (xy).(xz) x.zx.y and x.yx.z(xz).(xy)
• Consensus
• (x.y)(y.z)(x.z)x.yxz and (xy).(yz).(xz)
  (xy).(xz)

4
Proving theorems

• Prove x.yx.yx
• distributivity x.yx.y x.(yy)
• complementarity x.(yy) x.(1)
• identity x.(1) x
• Prove xx.y x
• identity xx.y x.1x.y
• distributivity x.1x.y x.(1y)
• identity x.(1y) x.(1)
• identity x.(1) x
• NOR is equivalent of AND
• (xy) x.y
• NAND is equivalent of OR
• (x.y) xy

5
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 (xxx)
• Cout a.b.cin a.b.cin 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
  a.b.cin a.b.cin
• Then use distributivity to combine terms
• Cout (aa).b.cin (bb).a.cin
  a.b.(cincin)
• Next use complementarity to reduce
• Cout (1).b.cin (1).a.cin a.b.(1)
• Finally using identity gives Cout b.cin a.cin
  a.b

6
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