Sum-of-Products (Functions to Circuits)

Context: FIT1047_MOC · truth table ➔ expression ➔ gate circuit, systematically · output feeds Karnaugh Maps for minimisation · gates from Transistors and Logic Gates

Quick Revision

• 🎯 Objective: read the rows where $F=1$ ➔ one AND-product per row (complement the 0-variables) ➔ OR the products together ➔ build the circuit gate-for-gate.
• ⚡ Key Constraint: SOP is correct but usually NOT minimal — simplification (Karnaugh Maps) comes after extraction.

📝 Core

• Truth table = the function ➔ a truth table uniquely defines a Boolean function; SOP answers “how do I get an expression back out?”
• Product (minterm) ➔ for each row with $F=1$, AND all variables, overlining those that are $0$ in that row: row $X=0,Y=1,Z=0\Rightarrow \overline{X}Y\overline{Z}$.
• Sum ➔ OR every product: the expression is TRUE exactly on the selected rows — by construction it reproduces the table.
• Circuit recipe ➔ one NOT per complemented input ➔ one AND gate per product ➔ one OR gate collecting all products; fully mechanical.
• Equivalence ➔ different-looking expressions from the same table are equivalent; verify via Boolean Algebra Laws or the table itself.

📊 Applied Exercise — lecture table

Rows where $F(X,Y,Z)=1$: $010,011,100,110,111$.

$$\begin{array}{rl}F(X,Y,Z)& =\overline{X}Y\overline{Z}+\overline{X}YZ+X\overline{Y}\overline{Z}+XY\overline{Z}+XYZ\\ & \text{(5 products ⟹ 5 AND gates + 1 five-input OR — correct, not minimal)}\end{array}$$

Final Extracted Output: valid SOP; Karnaugh Maps reduces the same table to $Y+X\overline{Z}$.

✍️ Practice

$G(A,B)$ is $1$ exactly when the inputs differ (rows $01$ and $10$). Extract the SOP and name the function.

Answer

• $G=\overline{A}B+A\overline{B}$ — this is XOR (Exclusive-or).
• Circuit: two NOTs, two 2-input ANDs, one OR.
• Key move: complement exactly the variables that read $0$ in each selected row.

⚠️ Common Mistakes

• 💡 Products come only from 1-rows ➔ using 0-rows builds the complement $\overline{F}$ (that’s the product-of-sums path instead).
• 💡 Complement the 0s, not the 1s ➔ the product must evaluate to 1 on its row; a bar on the wrong variable kills the row.