To solve this problem with an adequate demonstration of the Quine McCluskey method, begin with the definition of this method, the method steps, and fully assess the quantities given to us.
Definition of the McCluskey Algorithm: A simpler method of Karnaugh mapping that adapts best for computer programming, and in some cases, pragmatics theory.
Step 1: Find the Prime Implicants:
First, rewrite the equation to make it visually simpler to solve to correctly identify the complement members of each variable. Without this step, finding the prime implicants can be slowed and you waste time trying to fix preventable errors by casual misidentification of which variables are complements and the variables that are normal.
S(A,B,C,D,E) = A’B’C’D’E + A’B’CD’E’ + A’B’CD’E + A’B’CDE + AB’C’D’E + ABCD’E
should be rewritten with the complement symbol here ' substituted to have bar above the complement variable letter to make A-bar, and so on.
Next, find the minterms for the members of the above expression for S.
With 5 variables, we have a possible 32 minterms because 25 = 32.
This makes the minterm expression in long form look like this if all 32 possibilities are cited in the expression of S.
S(A,B,C,D,E) = ∑(m0 + m1 + m2 + m3 + ... + m31)
As reference, here is the list of variable expressions per minterm for you to pick out the correct ones for the next steps.
Minterm Variable expression
m0 A'B'C'D'E'
m1 A'B'C'D'E
m2 A'B'C'DE'
m3 A'B'C'DE
m4 A'B'CD'E'
m5 A'B'CD'E
m6 A'B'CDE'
m7 A'B'CDE
m8 A'BC'D'E'
m9 A'BC'D'E
m10 A'BC'DE'
m11 A'BC'DE
m12 A'BCD'E'
m13 A'BCD'E
m14 A'BCDE'
m15 A'BCDE
m16 AB'C'D'E'
m17 AB'C'D'E
m18 AB'C'DE'
m19 AB'C'DE
m20 AB'CD'E'
m21 AB'CD'E
m22 AB'CDE'
m23 AB'CDE
m24 ABC'D'E'
m25 ABC'D'E
m26 ABC'DE'
m27 ABC'DE
m28 ABCD'E'
m29 ABCD'E
m30 ABCDE'
m31 ABCDE
Luckily, we only have six minterm members of S to solve.
Thus, expect the minterm expression of S to look like:
S = m# + m# + m# + m# + m# + m#
Convert each minterm member of the expression of S to its binary equivalent. (For simplicity, use 0 for complements and 1 for normal variables as standard practice.)
Examples like, m0 = 00000, m31 = 11111.
Make note of the amount of 1s each minterm has per binary form.
Assemble the minterms into common groups based on the amount of 1s per minterm.
Group 1 for the minterm members whose binary expression has one 1.
Group 2 for the minterm members whose binary expression has two 1s.
and so on until you have grouped all 6 of your minterms.
For example, if a minterm has five 1s, while the rest don't, then it exists alone in Group 5.
Step 2: First level pairing
Examine the groups you have made at the end of the last step. Now is the time you get to check between groups to see where it is possible to combine the minterms that only differ in notation by one bit. We note the position where there is a difference in binary notation in combined form, also indicating difference in bit depending on the position where the binary digit changed. These minterms are then acknowledged as pairs and we indicate those are used combinations. If there remain unpaired minterms, we note them later after Step 3.
If done correctly, we should see something like this -0000, 0-000, 00-00, 000-0, or 0000-. The end goal is to reduce redundant members of the expression, by getting rid of extra A, B, C, D, or E.
Step 3: Second Level Pairing
With the specific pairs of minterms found in Step 2, we try to combine these noted binary groupings to find if these pairs contrast by only 1 bit in the same binary position.
If so, we get to have fewer members to document as Prime Implicants. If no further combination pairings can occur because the difference is more than one bit, then the pairs from Step 2 become Prime Implicants by default.
Step 4: Prime Implicant Chart to find the Essential Prime Implicants
Now we are satisfied with the remaining Prime Implicants from Steps 2 & 3, we can create a chart where the columns are the minterms listed in numerical order from left to right, featuring rows that indicate the noted binary position differences for the Prime Implicants. Each Prime Implicant is indicated as a check marking the binary difference position at the relevant minterm column.
Review the chart when you finish checkmarking to find the checkmarks that only have 1 binary position difference to match each relevant minterm. If they only match one minterm, these are your essential prime implicants. All the Prime Implicants should be documented.
Step 5: Accounting for unpaired minterms
Check back on the unpaired minterms from Steps 2 and/or 3, and find out if the documented options of binary position differences can account for the difference in binary position for the unpaired minterms, If so, then you have covered everything. If not, only then is it permissible to get additional prime implicants from the chart to appropriately note that all minterms are covered correctly within the Quine-McClusky method. If a Prime Implicant is missing from covering the minterms it needs to, look for other existing implicants to cover the minterms, even if it means finding a new set of Prime Implicants. The chart is only meant to indicate the minimal possible set of variable expressions for S.
Step 6: Creating the Simplified SOP Expression
Now you have the assortment of four binary digits and one dash binary Prime Implicants to convert back to variable expressions.
Here are two general examples for a 5 variable problem:
0 1 0 - 0 => A' = 0, B = 1, C' = 0, D = -, E' = 0 => A'BC'E'
1 - 0 0 1 => A = 1, B = -, C' = 0, D' = 0, E = 1 => AC'D'E
Desired final form example: Simplified SOP
S(A,B,C,D,E) = A'BC'E +...+ AC'D'E
Note: Good luck OP, I am happy to meet if further clarification is needed, but a lot of this requires careful organized documenting, correctly constructed tables, and being willing to pause and contemplate the mind-shift between notations. None of this is intended to align with human intuition. Logic acts as a machine's intuition, which is incredibly different.
It is rare to require five variable minterm expressions on anything other than homework in university-level logic design.
