Theory
Why engineers shrink formulas
Two circuits do the same job. One uses 5 gates, the other 2. The 2-gate version is cheaper, faster and cooler, literally: fewer gates burn less power.
Both came from the same Boolean function; someone just minimized the expression before wiring it.
That is this topic: standard ways to write a function, and laws to shrink it without changing what it does.
Theory
Different routes, same destination
Expressions are routes; the function is the destination. A·B + A·B·C and plain A·B deliver identical outputs for every input, like a winding road and a highway ending at the same office. Minimization is choosing the highway. The truth table (the destination) never changes.
Theory
The two standard costumes: SOP and POS
- SOP (sum of products): OR of AND terms, like A·B + ¬A·C. The canonical version uses minterms: every term contains all variables, one term per output-1 row.
- POS (product of sums): AND of OR terms, like (A + B)(¬A + C). Canonical POS uses maxterms, one per output-0 row.
Rule of thumb: few 1s in the table → SOP is shorter; few 0s → POS is shorter.
Quiz
Which of these is in SOP (sum of products) form?
- (A + B)·(A + C)
- A·B + ¬A·C
- ¬(A·B + C)
- A + (B·(C + A))
Show the answer
A·B + ¬A·C
SOP means AND terms joined by OR at the top level: A·B + ¬A·C fits exactly. Option A is POS (OR terms joined by AND). Option C has a complement wrapped around everything, and option D nests a sum inside a product, so neither is a standard form.
Theory
Worked minimization, every law named
Simplify F = A·B + A·B·C.
1. Factor the common A·B (distributive law): F = A·B(1 + C)
2. Domination: 1 + C = 1
3. Identity: A·B·1 = A·B
So F = A·B. The C term was pure decoration: whenever A·B·C is 1, A·B was already 1. This pattern, X + X·Y = X, is so common it has its own name: the absorption law.
Think first
Simplify G = (A + B)(A + ¬B) using the laws.
Try the distribution in your head first: multiply it out, kill the impossible term, then tap.
Show the answer
Multiply out (distributive): G = A·A + A·¬B + B·A + B·¬B
= A + A·¬B + A·B + 0 (idempotent A·A = A; complement B·¬B = 0)
= A(1 + ¬B + B) = A·1 = A
Shortcut worth memorising: (X + Y)(X + ¬Y) always collapses to X. The Y part cancels itself.
Theory
The minimizer's toolbox
Six laws do nearly all BCA-level minimization:
- Idempotent: A + A = A, A·A = A
- Complement: A + ¬A = 1, A·¬A = 0
- Identity / domination: A + 0 = A, A·1 = A, A + 1 = 1, A·0 = 0
- Distributive: factor or expand
- Absorption: A + A·B = A
- De Morgan's: ¬(A·B) = ¬A + ¬B and ¬(A + B) = ¬A·¬B, for moving complements inside
Name the law at each step; examiners award marks per named step.
Quiz
Simplify F = X + X·Y + X·Z.
- X·(Y + Z)
- X + Y + Z
- X
- X·Y·Z
Show the answer
X
Absorption twice: X + X·Y = X, then X + X·Z = X. Whenever X·Y or X·Z is 1, X itself is already 1, so the extra terms add nothing. Option A forgot the lone X term, and option B wrongly promotes Y and Z to independent conditions.
Watch out
The two silent killers
First: distributing complements without flipping the operation. ¬(A·B) is ¬A + ¬B, never ¬A·¬B; De Morgan's demands the flip. Second: "simplifying" into a different function. When unsure, spot-check one input row against the original expression; equivalent expressions must agree on every row.
Theory
Where you will meet this again
Compilers minimize your if-conditions with these exact laws, and later semesters hand you the Karnaugh map, a visual grid that does this minimization by eye. The next topic applies today's toolbox end to end: designing a real circuit from a sentence.
Summary
Key takeaways
- SOP is an OR of AND terms (minterms cover the 1-rows); POS is an AND of OR terms (maxterms cover the 0-rows).
- Minimization shrinks expressions without changing the truth table: fewer literals, fewer gates.
- Core toolbox: idempotent, complement, identity, domination, distributive, absorption, De Morgan's.
- Reusable shortcuts: X + X·Y = X and (X + Y)(X + ¬Y) = X.
- Memory hook: same table, shorter formula.