PRASHNIKAप्रश्निका

Boolean algebra solver

Simplify a Boolean expression the way ISC Computer Science asks for it: the working law by law, the truth table, the K-map with its groups, the minimal SOP and POS forms, and the logic circuit - also with NAND or NOR gates only. Free, with nothing to sign up for.

Check that it equals another expression

Write NOT as A' (or ~A), AND as AB or A.B, OR as A + B; ⊕, => and <=> work too. Up to 6 variables. Or give minterms: F(A,B,C) = Σ(1,3,5) + d(7), or maxterms with π.

Try: A'B'C + A'BC + AB'C + ABC' + ABCF(A,B,C,D) = Σ(0,2,5,7,8,10,13,15)F(P,Q,R,S) = π(0,1,3,5,7,8,9,11) + d(2,13)(X + Z)(XY + YZ) + XZ + Y(A + B')(A' + B)A ⊕ B ⊕ C

Answer

Minimal SOP
F = AB + C
Minimal POS
F = (A + C)(B + C)
Minterms
F(A, B, C) = Σ(1, 3, 5, 6, 7)
Maxterms
F(A, B, C) = π(0, 2, 4)

Read as A'B'C + A'BC + AB'C + ABC' + ABC. It is a contingency (sometimes 1, sometimes 0).

Steps, law by law

F = A'B'C + A'BC + AB'C + ABC' + ABC

  1. = A'C(B' + B) + AB'C + ABC' + ABCDistributive law AB + AB' = A(B + B')
  2. = A'.C.1 + AB'C + ABC' + ABCComplement law A + A' = 1
  3. = A'C + AB'C + ABC' + ABCIdentity law A.1 = A
  4. = A'C + AC(B' + B) + ABC'Distributive law AB + AB' = A(B + B')
  5. = A'C + A.C.1 + ABC'Complement law A + A' = 1
  6. = A'C + AC + ABC'Identity law A.1 = A
  7. = C(A' + A) + ABC'Distributive law AB + AB' = A(B + B')
  8. = C.1 + ABC'Complement law A + A' = 1
  9. = C + ABC'Identity law A.1 = A
  10. = C + ABAbsorption law A + A'B = A + B
Convert to POS, step by step

Work out F' as a sum of products, then F = (F')' by De Morgan's law.

  1. F' = (C + AB)'Complement F' = (F)'
  2. F' = C'(AB)'De Morgan's law (A + B)' = A'B'
  3. F' = C'(A' + B')De Morgan's law (AB)' = A' + B'
  4. F' = C'A' + C'B'Distributive law A(B + C) = AB + AC
  5. F = (C'A' + C'B')'Complement again F = (F')'
  6. F = (C'A')'(C'B')'De Morgan's law (A + B)' = A'B'
  7. F = ((C')' + (A')')(C'B')'De Morgan's law (AB)' = A' + B'
  8. F = (C + (A')')(C'B')'Involution law (A')' = A
  9. F = (C + A)(C'B')'Involution law (A')' = A
  10. F = (C + A)((C')' + (B')')De Morgan's law (AB)' = A' + B'
  11. F = (C + A)(C + (B')')Involution law (A')' = A
  12. F = (C + A)(C + B)Involution law (A')' = A

Truth table

ABCAB'CA'B'C + A'BC + AB'CABC'ABC'A'B'C + A'BC + AB'C + ABC'ABCA'B'C + A'BC + AB'C + ABC' + ABC
00000010000
00101000101
01000010000
01101000101
10000010000
10111000101
11000111101
11100100011

K-map

Grouping the 1s (SOP)
BCA00011110010113124517161
  • AB pair: cells 6, 7
  • C quad: cells 1, 3, 5, 7
Grouping the 0s (POS)
BCA000111100100132040576
  • (A + C) pair: cells 0, 2
  • (B + C) pair: cells 0, 4

Canonical forms

SOP (sum of minterms)

F = A'B'C + A'BC + AB'C + ABC' + ABC

POS (product of maxterms)

F = (A + B + C)(A + B' + C)(A' + B + C)

Logic circuit

Your expression as written Gate for gate as typed; A' is taken as an input line.
Logic circuit: Your expression as written
Minimal SOP AND, OR and NOT gates.
Logic circuit: Minimal SOP
Minimal POS OR, AND and NOT gates.
Logic circuit: Minimal POS
NAND gates only From the minimal SOP; a NOT is a NAND with its inputs joined.
Logic circuit: NAND gates only
NOR gates only From the minimal POS; a NOT is a NOR with its inputs joined.
Logic circuit: NOR gates only

Practise on real ISC questions