Trace and simplify logic circuits

Signals, inputs and outputs

Use ideal Boolean signals: 0 means false and 1 true. Each gate computes its output from its inputs. These are combinational circuits: there is no feedback, stored state, voltage calculation or propagation-delay model. The output is determined by the current input assignment

Read the labelled signal-flow diagrams

The circles here label primary inputs or gate outputs. The accompanying gate description gives the operation and each input port. Arrows run from a signal source to the gate that receives it. F names the final output. This is a labelled schematic, not a drawing of conventional gate outlines or an electrical wiring plan

AND, inclusive OR and NOT

A two-input AND gives 1 only when both inputs are 1. A two-input OR gives 1 when at least one is 1, including both. NOT has one input and reverses it. The Boolean notation from 3B writes AND as a product, OR as a sum and NOT with a bar

NAND and NOR complement a whole result

NAND means NOT of AND; NOR means NOT of inclusive OR. Complement the combined result, not just one chosen input. In conventional gate notation an output inversion bubble indicates this complement; the labelled diagrams here use the words NAND or NOR

F is NAND(P,Q). With P = 1 and Q = 1, what is F?

With P = 1 and Q = 0, which pair gives NAND(P,Q), then NOR(P,Q)?

Wires copy signals; they do not apply OR

A signal may feed several gate ports without changing its value. Two lines that cross away from a named endpoint do not create a connection. Never join two different gate outputs as if the junction were an OR gate. If both ports receive the same named input, they are tied inputs, not independent variables

Both ports of NOR gate F receive the same P. If P = 1, what is F?

Trace in dependency order

Start with the primary input values. Compute a gate only after all of its input signals are known. Write each intermediate output before using it downstream. This is a topological order of the directed acyclic circuit. A loop would need a different model and is excluded here

X = NOT(P), then F = AND(X,Q). For P = 0 and Q = 1, what are X and F?

Translate a circuit into an expression

Name intermediate outputs, then substitute them into later gate equations. Preserve which operation each complement covers. Substitute the NOT output into the AND gate as shown below. To go from an expression to a circuit, build its inner operations first and connect their outputs to the outer operation

Construct a circuit, ungraded

Construct F = (P + Q)(not R) using two-input AND and OR gates and one-input NOT gates. Name every intermediate output and list the source feeding every port. Draw arrows and verify all eight input assignments. This independent drawing and verification are ungraded

Model: a three-gate construction

Use X = OR(P,Q), Y = NOT(R), then F = AND(X,Y). Branching is not needed. Substitute to obtain F = (P + Q)(not R). In input order 000,001,010,011,100,101,110,111, the final values are 0,0,1,0,1,0,1,0. A different drawing with the same gate operations and connections is equivalent

Inspect the model: X = OR(P,Q), Y = NOT(R), F = AND(X,Y). With P = 0, Q = 1 and R = 1, what is F?

Equivalence must hold for every input

Two circuits are equivalent when they produce the same final output for every assignment of the same primary inputs. Their intermediate signals and number of gates may differ. One matching trial proves only that trial. For two inputs check all four assignments; for three check all eight, or use a valid Boolean identity

NAND alone can build the basic operations

Tying both NAND inputs to P gives NOT(P). Invert a NAND output with another tied-input NAND to recover AND. To build OR, first invert each input using a tied-input NAND, then NAND those two outputs. These constructions show that combinations of NAND can implement any circuit formed from NOT, AND and OR

Construct with NAND only, ungraded

Build inclusive OR of P and Q using only two-input NAND gates. Specify all ports, including any tied inputs. Prove equivalence with De Morgan and all four input assignments. Independent circuit construction and proof are ungraded

Model: three NAND gates

Set X = NAND(P,P), Y = NAND(Q,Q), F = NAND(X,Y). Thus X = not P, Y = not Q, and F is NOT of their AND, equal to P + Q. For input rows 00,01,10,11, the triples (X,Y,F) are (1,1,0), (1,0,1), (0,1,1), (0,0,1). The final column is exactly inclusive OR

X = NAND(P,P), Y = NAND(Q,Q), F = NAND(X,Y). Which operation does this entire circuit perform?

NOR alone can also build the basic operations

A tied-input NOR gives NOT. A NOR followed by a tied-input NOR gives OR. NOR of separately inverted inputs gives AND. The same universality argument applies: replace each NOT, AND or OR by its NOR-only construction. Universal does not mean one gate performs every function

Simplification and gate counts need assumptions

Simplifying the Boolean expression can reduce a circuit while preserving its truth function. Count gates, not wires or labels. Here AND, OR, NAND and NOR have two inputs, NOT has one, a wire may branch, and each gate counts once. A smaller gate count does not by itself establish lower cost or delay in real hardware, and it is not a proof of global optimality

Simplify and redraw, ungraded

A circuit uses X = OR(P,Q), Y = OR(P,R), F = AND(X,Y). Simplify its expression and construct an equivalent circuit using fewer basic gates counting each two-input AND or OR gate once; wires and signal labels do not count. Show a Boolean derivation and verify every input assignment. Independent redesign and justification are ungraded

Model: three gates become two

The original has three gates and computes (P + Q)(P + R). Expand the product, then use absorption to remove the two terms containing P. Use one AND for T = QR and one OR for F = P + T. Both produce 0,0,0,1,1,1,1,1 in input order 000 through 111. This proves equivalence and a reduction from three gates to two, without claiming a universal optimisation rule

With T = AND(Q,R) and F = OR(P,T), how many gates are used? Count each listed gate once

Diagnose a changed gate, ungraded

A correct circuit has X = NAND(P,Q), followed by F = NOT(X). Someone replaces the NAND with NOR but leaves the final NOT. Find one input assignment that distinguishes the two circuits, explain the changed function, and check all four rows. This independent diagnosis is ungraded

Model: a distinguishing input

The original computes PQ because the final NOT cancels the NAND complement. The changed circuit computes P + Q because the final NOT now cancels a NOR complement. At P = 0, Q = 1, the original gives 0 and the changed circuit 1. Across 00,01,10,11 their columns are 0,0,0,1 and 0,1,1,1. Agreement at 00 or 11 alone would miss the fault

Simulate the circuit, then verify the model

In a spreadsheet or short program, enumerate the two, four or eight assignments to the original inputs. For each row, evaluate gates in dependency order and record each named output. Compare the final column with a separately evaluated expression. Use Boolean operations, not ordinary addition for OR. Repeated ports must read the same stored signal. This exhaustive simulation verifies the finite truth function; it does not model voltages, delay or feedback

Diagram-to-expression task (ungraded): X is NAND, Y is AND, Z is NOT and F is OR. Use the arrows to write an equation for each gate, then express F using P, Q and R only. Construct a complete truth table and simplify F. Keep this written work for self-review. How many input assignments must your table check?

Model: read connections before simplifying

X receives P and Q; NAND complements their product. Y receives P and R. Z reverses X, and F combines Y and Z with inclusive OR. Substitute the intermediate expressions without changing the scope of either complement

Model: every input assignment

X is NAND of P and Q, Y is AND of P and R, Z is NOT of X, and F is OR of Y and Z. List the input rows in binary order. Work across each row in gate dependency order. X and Z are opposite on every row. In the last row, OR of 1 and 1 is 1, not the arithmetic sum 2.

Model: simplify and check all rows

The final column is 1 exactly when P is 1 and at least one of Q and R is 1. Factor P using the Boolean distributive law. A two-gate replacement uses T = OR(Q,R), then F = AND(P,T). It agrees on all eight rows and uses fewer gates than the four-gate original; this comparison does not claim global optimality

Self-review the complete response

The circuit has P and Q feeding NAND gate X, P and R feeding AND gate Y, X feeding NOT gate Z, and Y and Z feeding OR gate F. Check that your equations match these connections, each complement covers the correct expression, all eight input rows occur once, and every intermediate column is labelled. Justify the simpler rule using its full truth column or Boolean laws. For an independent technology check, enter the gate operations in separate spreadsheet columns and compare the final column with P AND (Q OR R); keep your formulas and results. The written response and technology record remain ungraded