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.