Boolean Algebra - Binary logic

Keep in mind, this is not the only order in which these can be simplified.





1.  !(A*!B+!A*B)=A*B+!(A+B)

 !(A*!B+!A*B)
 deMorgan
 !(A*!B)*!(!A*B)
 deMorgan
 (!A+!!B)*(!!A*!B)
 Cancellation
 (!A+B)(A*!B)
 Distribution
 !A*A+A*B+!A*!B+B*!B
 Inverse
 0+A*B+!A*!B+0
 Null
 A*B+!A*!B
 deMorgan
 A*B+!(A+B)



2. (A+B)*(A+C)*(!B+!C)=A(!B+!C)

 (A+B)*(A+C)*(!B+!C)=A(!B+!C)
 Ditribution - on (A+B)*(A+C) 
 (A+B*C)*(!B+!C)
 Distribution
 A*!B+B*C*!B+A*!C+B*C*!C
 Inverse
 A*!B+0*C+A*!C+B*0
 Null
 A*!B+A*!C
 Distribution
 A*(!B+!C)






3. !(!(!(A*B)*!(!(A*A)*!(B*B))))=A*!B+!A*B

 !(!(!(A*B)*!(!(A*A)*!(B*B))))
 Itempotent
 !(!(!(A*B)*!(!A*!B)))
 deMorgan
 !!(!(A*B)+!!(!A*!B)) 
 Inverse
 !(A*B)+(!A*!B)
 deMorgan
 (!A+!B)*!(!A*!B)
 deMorgan
 (!A+!B)*(!!A+!!B)
 Cancellation
 (!A+!B)*(A+B)
 Distribution
 !A*A+!B*A+!A*B+!B*B
 Inverse
 0+!B*A+!A*B+0
 Null
 !B*A+!A*B



4. !(!(!(!(A+A)+B)+!(A+!(B+B))))=A*!B+!A*B

 !(!(!(!(A+A)+B)+!(A+!(B+B))))
 Cancellation
 !(!(A+A)+B)+!(A+!(B+B))
 Itempotent 
 !(!A+B)+!(A+!B)
 deMorgan
 !!A*!B+!A*!!B
 Inverse
 A*!B+!A*B






5. A*(B+C)+!A*B*C=A*B+A*C+B*C

 A*(B+C)+!A*B*C
 Distribution
 A*B+A*C+!A*B*C
 Distribution 
 A*B+(A+!A*B)*C
 --------------      
      x+yz = (x+y)(x+z)
 --------------      
 Distribution
 A*B+(A+!A)*(A+B)*C
 Inverse
 A*B+(1)*(A+B)*C
 Null
 A*B+(A+B)*C
 Distribution
 A*B+A*C+B*C