1. a · a = 0
  2. a + a = 1
  3. ab = ba
  4. a+b = b+a
  5. (a+b)+c = a + (b + c)
  6. (ab)c = a(bc)
  7. ab + ac = a(b+c)
  8. a(b+c) = ab + ac
  9. aa = a
  10. a+a = a
  11. a+b = a · b
  12. a·b = a + b
  13. a·1 = a
  14. a + 1 = 1
  15. a· 0 = 0
  16. a + 0 = a

Some Proofs

  • With the rules we have learned, we can prove some useful facts.
  • We can always try to prove with a truth table.
  • But sometimes manipulating with the boolean rules is helpful.
  • Show a+ab = a 
    • a+ab = a(1+b)
     = a(1)
     = a
  • a + ab = a + b 
    •  a + ab =  a(b + b) + ab
        =  ab + ab + ab
        =  ab + ab + ab + ab
        =  ab + ab + ab + ab
        =  a(b + b) + b(a + a)
        = a(1) + b(1)
        = a b
  • a(a+b) = a 
    • a(a+b) = aa + ab
       = a + ab 
       = a (first proof)
  • (a+b)(a+b) = a 
    • (a+b)(a+b) = aa + ab + ab +  bb
           = a + ab + ab + 0 
           = a + a(b + b) 
           = a + a(1)
           = a + a 
           = a
  • We may do some more of these as we need them.