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 = XZ + Y
Minimal POS
F = (X + Y)(Y + Z)
Minterms
F(X, Y, Z) = Σ(2, 3, 5, 6, 7)
Maxterms
F(X, Y, Z) = π(0, 1, 4)

Read as (X + Z)(XY + YZ) + XZ + Y. It is a contingency (sometimes 1, sometimes 0).

Steps, law by law

F = (X + Z)(XY + YZ) + XZ + Y

  1. = X(XY + YZ) + Z(XY + YZ) + XZ + YDistributive law A(B + C) = AB + AC
  2. = XXY + XYZ + Z(XY + YZ) + XZ + YDistributive law A(B + C) = AB + AC
  3. = XY + XYZ + Z(XY + YZ) + XZ + YIdempotent law A.A = A
  4. = XY + Z(XY + YZ) + XZ + YAbsorption law A + AB = A
  5. = Z(XY + YZ) + XZ + YAbsorption law A + AB = A
  6. = ZXY + ZYZ + XZ + YDistributive law A(B + C) = AB + AC
  7. = ZXY + ZY + XZ + YIdempotent law A.A = A
  8. = ZY + XZ + YAbsorption law A + AB = A
  9. = XZ + YAbsorption law A + AB = A
Convert to POS, step by step

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

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

Truth table

XYZX + ZXYYZXY + YZ(X + Z)(XY + YZ)XZ(X + Z)(XY + YZ) + XZ(X + Z)(XY + YZ) + XZ + Y
00000000000
00110000000
01000000001
01110111011
10010000000
10110000111
11011011011
11111111111

K-map

Grouping the 1s (SOP)
YZX00011110010131214517161
  • XZ pair: cells 5, 7
  • Y quad: cells 2, 3, 6, 7
Grouping the 0s (POS)
YZX000111100100103240576
  • (X + Y) pair: cells 0, 1
  • (Y + Z) pair: cells 0, 4

Canonical forms

SOP (sum of minterms)

F = X'YZ' + X'YZ + XY'Z + XYZ' + XYZ

POS (product of maxterms)

F = (X + Y + Z)(X + Y + Z')(X' + Y + Z)

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