Simplify the Boolean expression F = A'B'C'D' + A'B'CD' + A'BC'D + A'BCD' + AB'C'D' + AB'C'D + ABC'D' + ABCD using the laws of Boolean algebra.

Step-by-step solution

1. 1

   List the minterms before touching the algebra. Reading $ABCD$ as a 4-bit number (with $A$ the most significant bit and a primed literal meaning $0$), the eight products are

   $m_0,\ m_2,\ m_5,\ m_6,\ m_8,\ m_9,\ m_{12},\ m_{15}$

   Having the index list makes it possible to verify the final answer instead of trusting the manipulation.

2. 2

   Combine pairs that differ in exactly one literal. The only rule doing real work is $X + X' = 1$ applied after factoring. For example

   $A'B'C'D' + A'B'CD' = A'B'D'(C'+C) = A'B'D'$

   and

   $AB'C'D' + AB'C'D = AB'C'(D'+D) = AB'C'$

3. 3

   Reuse minterms — this is the step that is usually skipped. Idempotence, $X + X = X$, lets a minterm appear in more than one grouping at no cost. Duplicating $m_6$ and $m_{12}$ produces two more three-literal terms:

   $A'BCD' + A'B'CD' = A'CD'(B+B') = A'CD'$
   $ABC'D' + AB'C'D' = AC'D'(B+B') = AC'D'$

   Stopping at the first pairing leaves $A'BCD'$ and $ABC'D'$ at four literals each, which is why the "obvious" grouping is not minimal.

4. 4

   Handle the two minterms with no neighbour. $m_5 = A'BC'D$ and $m_{15} = ABCD$ differ from every other listed minterm in more than one bit, so neither can be combined. They survive as essential prime implicants in full four-literal form.

5. 5

   Assemble the minimal sum of products.

   $F = A'B'D' + A'CD' + A'BC'D + AB'C' + AC'D' + ABCD$

   Six product terms, 20 literals in total. (An equally minimal alternative swaps $A'B'D'$ for $B'C'D'$; both cover $m_0$, so the minimum is not unique.)

6. 6

   Verify by exhaustive evaluation, not by inspection. Evaluating the simplified expression over all $16$ assignments of $(A,B,C,D)$ gives $1$ exactly on $\{0,2,5,6,8,9,12,15\}$ — the same set as the original. Every one of the eight original minterms is covered and no extra row turns on, so the two expressions are equal.