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

Read as A ⊕ B ⊕ C. It is a contingency (sometimes 1, sometimes 0).

Steps, law by law

F = (A ⊕ B) ⊕ C

  1. = (A ⊕ B).C' + (A ⊕ B)'CDefinition of XOR A ⊕ B = AB' + A'B
  2. = (AB' + A'B)C' + (A ⊕ B)'CDefinition of XOR A ⊕ B = AB' + A'B
  3. = (AB' + A'B)C' + (AB' + A'B)'CDefinition of XOR A ⊕ B = AB' + A'B
  4. = (AB' + A'B)C' + (AB')'(A'B)'CDe Morgan's law (A + B)' = A'B'
  5. = (AB' + A'B)C' + (A' + (B')')(A'B)'CDe Morgan's law (AB)' = A' + B'
  6. = (AB' + A'B)C' + (A' + B)(A'B)'CInvolution law (A')' = A
  7. = (AB' + A'B)C' + (A' + B)((A')' + B')CDe Morgan's law (AB)' = A' + B'
  8. = (AB' + A'B)C' + (A' + B)(A + B')CInvolution law (A')' = A
  9. = AB'C' + A'BC' + (A' + B)(A + B')CDistributive law A(B + C) = AB + AC
  10. = AB'C' + A'BC' + A'(A + B')C + B(A + B')CDistributive law A(B + C) = AB + AC
  11. = AB'C' + A'BC' + A'B'C + B(A + B')CAbsorption law A(A' + B) = AB
  12. = AB'C' + A'BC' + A'B'C + BACAbsorption law A(A' + B) = AB
Convert to POS, step by step

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

  1. F' = (AB'C' + A'BC' + A'B'C + BAC)'Complement F' = (F)'
  2. F' = (AB'C')'(A'BC')'(A'B'C)'(BAC)'De Morgan's law (A + B)' = A'B'
  3. F' = (A' + (B')' + (C')')(A'BC')'(A'B'C)'(BAC)'De Morgan's law (AB)' = A' + B'
  4. F' = (A' + B + (C')')(A'BC')'(A'B'C)'(BAC)'Involution law (A')' = A
  5. F' = (A' + B + C)(A'BC')'(A'B'C)'(BAC)'Involution law (A')' = A
  6. F' = (A' + B + C)((A')' + B' + (C')')(A'B'C)'(BAC)'De Morgan's law (AB)' = A' + B'
  7. F' = (A' + B + C)(A + B' + (C')')(A'B'C)'(BAC)'Involution law (A')' = A
  8. F' = (A' + B + C)(A + B' + C)(A'B'C)'(BAC)'Involution law (A')' = A
  9. F' = (A' + B + C)(A + B' + C)((A')' + (B')' + C')(BAC)'De Morgan's law (AB)' = A' + B'
  10. F' = (A' + B + C)(A + B' + C)(A + (B')' + C')(BAC)'Involution law (A')' = A
  11. F' = (A' + B + C)(A + B' + C)(A + B + C')(BAC)'Involution law (A')' = A
  12. F' = (A' + B + C)(A + B' + C)(A + B + C')(B' + A' + C')De Morgan's law (AB)' = A' + B'
  13. F' = A'(A + B' + C)(A + B + C')(B' + A' + C') + B(A + B' + C)(A + B + C')(B' + A' + C') + C(A + B' + C)(A + B + C')(B' + A' + C')Distributive law A(B + C) = AB + AC
  14. F' = A'(A + B' + C)(A + B + C') + B(A + B' + C)(A + B + C')(B' + A' + C') + C(A + B' + C)(A + B + C')(B' + A' + C')Absorption law A(A + B) = A
  15. F' = A'(A + B' + C)(A + B + C') + B(A + B' + C)(B' + A' + C') + C(A + B' + C)(A + B + C')(B' + A' + C')Absorption law A(A + B) = A
  16. F' = A'(A + B' + C)(A + B + C') + B(A + B' + C)(B' + A' + C') + C(A + B + C')(B' + A' + C')Absorption law A(A + B) = A
  17. F' = A'(B' + C)(A + B + C') + B(A + B' + C)(B' + A' + C') + C(A + B + C')(B' + A' + C')Absorption law A(A' + B) = AB
  18. F' = A'(B' + C)(B + C') + B(A + B' + C)(B' + A' + C') + C(A + B + C')(B' + A' + C')Absorption law A(A' + B) = AB
  19. F' = A'(B' + C)(B + C') + B(A + C)(B' + A' + C') + C(A + B + C')(B' + A' + C')Absorption law A(A' + B) = AB
  20. F' = A'(B' + C)(B + C') + B(A + C)(A' + C') + C(A + B + C')(B' + A' + C')Absorption law A(A' + B) = AB
  21. F' = A'(B' + C)(B + C') + B(A + C)(A' + C') + C(A + B)(B' + A' + C')Absorption law A(A' + B) = AB
  22. F' = A'(B' + C)(B + C') + B(A + C)(A' + C') + C(A + B)(B' + A')Absorption law A(A' + B) = AB
  23. F' = A'B'(B + C') + A'C(B + C') + B(A + C)(A' + C') + C(A + B)(B' + A')Distributive law A(B + C) = AB + AC
  24. F' = A'B'C' + A'C(B + C') + B(A + C)(A' + C') + C(A + B)(B' + A')Absorption law A(A' + B) = AB
  25. F' = A'B'C' + A'CB + B(A + C)(A' + C') + C(A + B)(B' + A')Absorption law A(A' + B) = AB
  26. F' = A'B'C' + A'CB + BA(A' + C') + BC(A' + C') + C(A + B)(B' + A')Distributive law A(B + C) = AB + AC
  27. F' = A'B'C' + A'CB + BAC' + BC(A' + C') + C(A + B)(B' + A')Absorption law A(A' + B) = AB
  28. F' = A'B'C' + A'CB + BAC' + BCA' + C(A + B)(B' + A')Absorption law A(A' + B) = AB
  29. F' = A'B'C' + A'CB + BAC' + C(A + B)(B' + A')Idempotent law A + A = A
  30. F' = A'B'C' + A'CB + BAC' + CA(B' + A') + CB(B' + A')Distributive law A(B + C) = AB + AC
  31. F' = A'B'C' + A'CB + BAC' + CAB' + CB(B' + A')Absorption law A(A' + B) = AB
  32. F' = A'B'C' + A'CB + BAC' + CAB' + CBA'Absorption law A(A' + B) = AB
  33. F' = A'B'C' + A'CB + BAC' + CAB'Idempotent law A + A = A
  34. F = (A'B'C' + A'CB + BAC' + CAB')'Complement again F = (F')'
  35. F = (A'B'C')'(A'CB)'(BAC')'(CAB')'De Morgan's law (A + B)' = A'B'
  36. F = ((A')' + (B')' + (C')')(A'CB)'(BAC')'(CAB')'De Morgan's law (AB)' = A' + B'
  37. F = (A + (B')' + (C')')(A'CB)'(BAC')'(CAB')'Involution law (A')' = A
  38. F = (A + B + (C')')(A'CB)'(BAC')'(CAB')'Involution law (A')' = A
  39. F = (A + B + C)(A'CB)'(BAC')'(CAB')'Involution law (A')' = A
  40. F = (A + B + C)((A')' + C' + B')(BAC')'(CAB')'De Morgan's law (AB)' = A' + B'
  41. F = (A + B + C)(A + C' + B')(BAC')'(CAB')'Involution law (A')' = A
  42. F = (A + B + C)(A + C' + B')(B' + A' + (C')')(CAB')'De Morgan's law (AB)' = A' + B'
  43. F = (A + B + C)(A + C' + B')(B' + A' + C)(CAB')'Involution law (A')' = A
  44. F = (A + B + C)(A + C' + B')(B' + A' + C)(C' + A' + (B')')De Morgan's law (AB)' = A' + B'
  45. F = (A + B + C)(A + C' + B')(B' + A' + C)(C' + A' + B)Involution law (A')' = A

Truth table

ABCA ⊕ BA ⊕ B ⊕ C
00000
00101
01011
01110
10011
10110
11000
11101

K-map

Grouping the 1s (SOP)
BCA0001111001011321415716
  • ABC single: cell 7
  • AB'C' single: cell 4
  • A'BC' single: cell 2
  • A'B'C single: cell 1
Grouping the 0s (POS)
BCA0001111001001302450760
  • (A' + B' + C) single: cell 6
  • (A' + B + C') single: cell 5
  • (A + B' + C') single: cell 3
  • (A + B + C) single: cell 0

Canonical forms

SOP (sum of minterms)

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

POS (product of maxterms)

F = (A + B + C)(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