Wednesday, October 9, 2013

Two's Complement Binary Addition

 -39 + 92 = 53:

1111
11011001
+01011100
00110101
Carryout without overflow. Sum is Correct.

104 + 45 = 149:
111
01101000
+00101101
10010101
Overflow, no carryout. Sum is not correct.

 10 + -3 = 7:
11111
00001010
+11111101
00000111
Carryout without overflow. Sum is correct.

-1 + 1 = 0:
11111111
11111111
+00000001
00000000
Carryout without overflow. Sum is correct.

Boolean Expression Simplification

  • Simplify: C + BC:
    ExpressionRule(s) Used
    C + BCOriginal Expression
    C + (B + C)DeMorgan's Law.
    (C + C) + BCommutative, Associative Laws.
    T + BComplement Law.
    TIdentity Law.
  • Simplify: AB(A + B)(B + B):
    ExpressionRule(s) Used
    AB(A + B)(B + B)Original Expression
    AB(A + B)Complement law, Identity law.
    (A + B)(A + B)DeMorgan's Law
    A + BBDistributive law. This step uses the fact that or distributes over and. It can look a bit strange since addition does not distribute over multiplication.
    AComplement, Identity.
  • Simplify: (A + C)(AD + AD) + AC + C:
    ExpressionRule(s) Used
    (A + C)(AD + AD) + AC + COriginal Expression
    (A + C)A(D + D) + AC + CDistributive.
    (A + C)A + AC + CComplement, Identity.
    A((A + C) + C) + CCommutative, Distributive.
    A(A + C) + CAssociative, Idempotent.
    AA + AC + CDistributive.
    A + (A + T)CIdempotent, Identity, Distributive.
    A + CIdentity, twice.
    You can also use distribution of or over and starting from A(A+C)+C to reach the same result by another route.
  • Simplify: A(A + B) + (B + AA)(A + B):
    ExpressionRule(s) Used
    A(A + B) + (B + AA)(A + B)Original Expression
    AA + AB + (B + A)A + (B + A)BIdempotent (AA to A), then Distributive, used twice.
    AB + (B + A)A + (B + A)BComplement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
    AB + BA + AA + BB + ABDistributive, two places.
    AB + BA + A + ABIdempotent (for the A's), then Complement and Identity to remove BB.
    AB + AB + AT + ABCommutative, Identity; setting up for the next step.
    AB + A(B + T + B)Distributive.
    AB + AIdentity, twice (depending how you count it).
    A + ABCommutative.
    (A + A)(A + B)Distributive.
    A + BComplement, Identity

Basic Logic Gates Problems

  • Draw a logic circuit for (A + B)C.
  • Draw a logic circuit for A + BC + D.
  • Draw a logic circuit for AB + AC.
  • Draw a logic circuit for (A + B)(C + D)C.


    Source : http://sandbox.mc.edu