Question 1

Boolean simplification — minimize F = ABD + CD + ACD + ABC + ABCD Using Boolean algebra (no K-map required), find the most simplified SOP form from the options provided.

Accepted Answer

Introduction / Context: Algebraic simplification reduces hardware by removing redundant terms. For the given four-variable function, we apply absorption and factoring to reach a minimal or near-minimal sum-of-products (SOP) that matches one of the choices. Given Data / Assumptions: F = ABD + CD + ACD + ABC + ABCD. Variables A, B, C, D are Boolean. Operators: + is OR, concatenation is AND. Concept / Approach: Use absorption: X + XY = X, and redundancy removal where a more specific product term is covered by a more general one. Combine terms where a factorization can eliminate literals without altering coverage of minterms. Step-by-Step Solution: Start: F = ABD + CD + ACD + ABC + ABCD.Observe ABCD is contained in ABD (and also in ACD and ABC). Apply absorption: remove ABCD.Now F = ABD + CD + ACD + ABC.Notice CD + ACD = CD(1 + A) = CD. Absorb ACD into CD.Thus F simplifies to F = ABD + ABC + CD. Verification / Alternative check: Factor the remaining two AB terms: F = AB(C + D) + CD. Re-expanding yields ABC + ABD + CD, identical to the simplified expression. A truth table or logic simulator would confirm equivalence with the original function across all 16 input combinations. Why Other Options Are Wrong: F = CD + AD: AD covers minterms where B = 0 that are not in the original, so it overstates the function. F = BC + AB: Omits required D-dependent coverage and includes minterms not present. F = AC + AD: Again introduces minterms not guaranteed by the original and misses specific AB-dependent ones. Common Pitfalls: Dropping CD entirely after absorbing ACD; CD must remain. Over-aggressive factoring leading to terms like AD or AB alone, which add unwanted minterms. Final Answer: F = ABD + ABC + CD

Question 2

K-map grouping rules — which cell combinations are NOT valid to group? In Karnaugh map minimization, identify the combination below that cannot be grouped into valid power-of-two implicants.

Accepted Answer

Introduction / Context: Karnaugh maps (K-maps) simplify Boolean expressions by grouping adjacent 1-cells (or 0-cells for POS) into power-of-two blocks. Understanding what counts as adjacency is crucial to form correct groups and achieve minimal logic. Given Data / Assumptions: Adjacency is defined between cells that differ by exactly one variable (Gray code ordering). Wrapping is allowed—edges and corners can be adjacent across the map boundaries. Groups must be 1, 2, 4, 8, … cells in rectangular shapes. Concept / Approach: Valid groupings include horizontally or vertically adjacent cells, with edge wrap making opposite edges (and therefore corners) adjacent. Overlapping groups are permitted when they enable larger implicants. Diagonal cells are not adjacent (they differ in two variables) and therefore cannot be grouped together as a pair by themselves. Step-by-Step Solution: Check “corners in the same row/column”: with wrap-around, corner cells can join to form a 2-cell or 4-cell group.Check “overlapping combinations”: allowed if each group size is a power of two and improves simplification.Check “diagonal”: diagonally opposite cells differ in two variables; not adjacent; cannot form a valid 2-cell group. Verification / Alternative check: Label a 4-variable K-map in Gray code. Compare coordinates of diagonal cells; Hamming distance is 2, confirming non-adjacency. Contrast with edge-adjacent cells where Hamming distance is 1, validating permitted groupings. Why Other Options Are Wrong: Corners in same row/column: valid due to wrap-around adjacency. Overlapping: permitted and often necessary to achieve larger implicants. Common Pitfalls: Forgetting edge wrap makes opposite edges adjacent. Forming L-shaped or diagonal-only groups, which violate the rectangular, power-of-two rule. Final Answer: diagonal

Question 3

Converting W(X + YZ) to standard SOP form By applying which Boolean algebra law can the expression W(X + YZ) be expanded into a sum-of-products (SOP) form?

Accepted Answer

Introduction / Context: In digital logic design, we frequently convert mixed expressions into a standard form for implementation with basic gates or programmable logic. The two most common canonical forms are sum-of-products (SOP) and product-of-sums (POS). This question asks which Boolean law lets us expand W(X + YZ) into SOP, where each term is a product (AND) of literals and the overall expression is a sum (OR) of those products. Given Data / Assumptions: The expression is W(X + YZ). Variables W, X, Y, Z are Boolean (take values 0 or 1). Goal: express as SOP (sum of product terms). Concept / Approach: The key law is Boolean distribution, which lets a factor multiply (AND) into a sum (OR). The distributive law appears in two useful forms: A(B + C) = AB + AC A + BC = (A + B)(A + C) (the dual, for POS work) Here, we need the first form to push W across the parentheses so that the result is a sum of product terms suitable for SOP implementation. Step-by-Step Solution: Start with: F = W(X + YZ)Apply distributive law: W(X + YZ) = WX + W(YZ)Associate the second product (implicit): W(YZ) = (WY)Z = WYZTherefore, F = WX + WYZ, which is clearly SOP (sum of the products WX and WYZ). Verification / Alternative check: You can construct a quick truth table for W, X, Y, Z and verify that F = 1 exactly when either WX = 1 or WYZ = 1. A Karnaugh map grouping would also show the same on-cells, confirming that distribution produced an equivalent SOP expression without changing functionality. Why Other Options Are Wrong: Associative law changes grouping only, such as (A + B) + C = A + (B + C); it does not expand across a sum. Commutative law swaps order, such as AB = BA or A + B = B + A; it does not expand products over sums. Idempotent law (A + A = A, AA = A) simplifies duplicate literals, not distribute factors. “None of the above” is incorrect because distribution is exactly what is used. Common Pitfalls: Stopping at WX + W(YZ) and forgetting that W(YZ) is already a valid product term WYZ; no further expansion is needed for SOP. Accidentally turning W + (YZ) into (W + Y)(W + Z), which is a POS transformation, not SOP. Final Answer: distributive law

Question 4

Interpreting De Morgan’s theorem Given the theorem (A * B)' = A' + B', which pair of gate descriptions are logically equivalent according to this statement?

Accepted Answer

Introduction / Context: De Morgan’s theorems are fundamental identities used to transform logic expressions and convert between gate types. They allow designers to swap between NAND/NOR implementations and AND/OR structures with input or output inversions, which is crucial for technology mapping and minimizing parts in practical circuits. Given Data / Assumptions: Use positive logic (1 = HIGH, 0 = LOW). De Morgan’s theorem provided: (A * B)' = A' + B'. Symbol * denotes logical AND; + denotes logical OR; ' denotes inversion. Concept / Approach: The given identity states that the complement of an AND equals the OR of the complements. In gate terms, an AND followed by an inverter (i.e., a NAND) is equivalent to an OR gate whose inputs are both inverted. This is the classic translation used to realize functions with only NAND gates, or to interpret bubble-logic schematics correctly. Step-by-Step Solution: Start with: Y = (A * B)'.Apply De Morgan: Y = A' + B'.Gate interpretation: left side is NAND; right side is an OR gate with inverted inputs (bubbles on A and B).Thus, a NAND gate has identical logic behavior to an OR gate with input inversions. Verification / Alternative check: Build a 2-input truth table for both expressions. For all four input combinations of A and B, (A * B)' and A' + B' match exactly. A quick logic simulator test or simple gate-level substitution confirms the equivalence in practical circuits. Why Other Options Are Wrong: NOR equals an AND with inverted inputs per the other De Morgan identity (A + B)' = A' * B', not the one given here. “AND and NOR with inverted inputs” mismatches the provided form. “NOR and NAND with inverted inputs” is not a direct De Morgan pair for this identity. “NAND and AND with inverted inputs” confuses complement placement; an AND with inverted inputs corresponds to a NOR under the other identity. Common Pitfalls: Confusing which De Morgan form you are using. Remember: complement of AND becomes OR of complements; complement of OR becomes AND of complements. Misreading bubble logic: bubbles on the inputs of an OR create a NAND-equivalent; bubbles on the inputs of an AND create a NOR-equivalent. Final Answer: a NAND and an OR gate with inverted inputs

Question 5

Cost Implications — Simplifying a combinational logic circuit generally increases the hardware cost. Evaluate the statement.

Accepted Answer

Introduction: Logic simplification aims to reduce the number of gates, inputs per gate, and interconnections while preserving function. Cost in digital hardware is closely tied to device count, input fan-in, and wiring complexity. This question tests whether simplification increases or decreases cost. Given Data / Assumptions: We compare an unsimplified vs simplified combinational circuit implementing the same Boolean function. Cost correlates with gate count, fan-in, and interconnect length. Concept / Approach: Using Boolean algebra or Karnaugh maps, simplification typically reduces literals and product/sum terms, which maps to fewer gates or gates with lower fan-in. Fewer resources normally imply lower silicon area and lower cost, not higher. Step-by-Step Solution: Step 1: Express function in canonical form (e.g., SOP or POS).Step 2: Apply Boolean algebra or K-map grouping to eliminate redundant literals.Step 3: Implement the minimized expression with fewer gates or smaller fan-in. Verification / Alternative check: Practical synthesis tools target reduced area and delay, confirming that simplification reduces hardware resources on average. Why Other Options Are Wrong: Correct: Contradicts the typical outcome; simplification reduces cost.Depends on technology only: While technology matters, simplification almost always helps across families.Insufficient information: Standard cost models are adequate to judge the direction. Common Pitfalls: Assuming more complex algebra means more hardware; the opposite is typical.Ignoring fan-in reduction benefits on delay and power. Final Answer: Incorrect

Question 6

De Morgan Application — When De Morgan s theorem is applied to a NOR function, can we obtain an equivalent expression that yields an identical truth table?

Accepted Answer

Introduction: De Morgan transformations are core tools for switching between sums with inversion and products with inverted variables. Understanding how NOR relates to AND with inverted inputs lets you re-express logic while preserving functionality. Given Data / Assumptions: NOR is defined as NOT (A + B + ...) We assume ideal Boolean algebra over binary variables Concept / Approach: De Morgan states: NOT (A + B) = (NOT A) (NOT B) and NOT (A B) = (NOT A) + (NOT B). Therefore, a NOR function is equivalent to an AND of complemented inputs. These two expressions generate identical truth tables because they are algebraically equivalent. Step-by-Step Solution: Step 1: Start with X = NOT (A + B).Step 2: Apply De Morgan: X = (NOT A) (NOT B).Step 3: Conclude NOR equals AND of inverted inputs; truth tables match exactly. Verification / Alternative check: Construct truth tables for both forms across all 4 input pairs (00, 01, 10, 11); outputs match in every row. Why Other Options Are Wrong: Incorrect: Violates standard Boolean identities.Only for two inputs / Only for three or more inputs: De Morgan laws apply for any number of inputs. Common Pitfalls: Confusing NOR with NAND relationships.Forgetting that distribution of NOT across operators flips OR to AND and vice versa. Final Answer: Correct

Question 7

Boolean Operations — Evaluate the statement: The OR function corresponds to Boolean multiplication and the AND function corresponds to Boolean addition.

Accepted Answer

Introduction: In Boolean algebra, operators map to addition and multiplication analogs, but not in the same way as everyday intuition might suggest. This question checks whether you remember which logic operation maps to which algebraic symbol. Given Data / Assumptions: Standard Boolean notation is used Symbol + denotes OR, and juxtaposition or * denotes AND Concept / Approach: By convention, OR corresponds to addition (+) and AND corresponds to multiplication (*). The statement reverses this mapping, so it is incorrect. Step-by-Step Solution: Step 1: Recall identities: A + 0 = A (OR identity) parallels addition identity.Step 2: Recall identities: A * 1 = A (AND identity) parallels multiplication identity.Step 3: Therefore, OR maps to + and AND maps to *; the prompt is reversed. Verification / Alternative check: Truth tables and algebraic properties (commutativity, associativity, distributivity) align with these mappings. Why Other Options Are Wrong: Correct: Contradicts standard Boolean convention.True only in arithmetic algebra: Boolean algebra is distinct; the mapping is definitional there.Depends on representation: Mainstream Boolean notation is standardized on these mappings. Common Pitfalls: Confusing symbolic similarities when switching between integer arithmetic and Boolean algebra.Assuming OR must be multiplication because it can increase value; not applicable in Boolean domain. Final Answer: Incorrect

Question 8

Design Tools — Evaluate the statement: CPLD design software can be used to create original circuits and experimentally verify Boolean rules and laws through simulation and implementation.

Accepted Answer

Introduction: Complex Programmable Logic Devices (CPLDs) are configured with vendor tools that support design entry, synthesis, simulation, and sometimes hardware programming. This workflow allows designers to validate Boolean identities by building circuits and observing results. Given Data / Assumptions: CPLD toolchains include schematic or HDL entry and simulation capabilities Verification can be performed via simulation waveforms or on real hardware after programming Concept / Approach: By writing testbenches and simulating truth tables, a designer can confirm Boolean rules (like De Morgan) across inputs. Implementation onto a CPLD enables live verification under real electrical conditions. Step-by-Step Solution: Step 1: Enter circuit using schematic or HDL.Step 2: Create a testbench to exhaustively stimulate inputs.Step 3: Run timing/functional simulation to observe outputs.Step 4: Program the CPLD to confirm behavior on hardware if needed. Verification / Alternative check: Cross-check simulation waveforms with expected algebraic truth tables; they should match. Why Other Options Are Wrong: Incorrect: Modern CPLD tools support such workflows.Only for FPGA devices, not CPLD: CPLDs also have vendor toolchains with simulation.Only for prebuilt IP cores: Users can design original logic, not just reuse IP. Common Pitfalls: Assuming CPLDs lack full design flow compared to FPGAs.Skipping testbench creation and relying only on ad hoc checks. Final Answer: Correct

Question 9

Boolean Laws — Commutative property in OR and AND: In what way does the order of operands affect the result when combining two variables?

Accepted Answer

Introduction: Commutativity is a bedrock law in Boolean algebra. It guarantees that operands can be swapped without altering the outcome for both OR and AND operations. This law simplifies algebraic manipulation and circuit realization. Given Data / Assumptions: Boolean domain with variables A and B Operators: OR (+) and AND (*) Concept / Approach: The commutative law states A + B = B + A and A * B = B * A. Because gate implementations also reflect these identities, physical circuits behave the same regardless of operand order at the input pins (assuming symmetric devices). Step-by-Step Solution: Step 1: Write identities: A + B = B + A and A * B = B * A.Step 2: Confirm via truth tables that each pair of expressions yields identical outputs for all input pairs.Step 3: Conclude that order does not affect results for OR and AND. Verification / Alternative check: Swap inputs on a two-input gate in a simulator; outputs remain unchanged across the entire input space. Why Other Options Are Wrong: Incorrect: Contradicts standard Boolean law.Correct only for OR / only for AND: Both operators are commutative. Common Pitfalls: Confusing commutativity with associativity or distributivity.Assuming physical pin labeling changes behavior; for symmetric gates it does not. Final Answer: Correct: Order does not change the result

Question 10

Minimization Method — The systematic reduction of logic expressions and circuits is accomplished using Boolean algebra and related techniques. Evaluate this statement.

Accepted Answer

Introduction: Simplifying logic reduces gate count, wiring, and power while preserving function. Boolean algebra provides a rigorous rule set for performing such reductions by hand, complemented by methods like Karnaugh maps and Quine McCluskey. Given Data / Assumptions: We are dealing with combinational logic expressions Goal is functional equivalence with fewer resources Concept / Approach: Boolean algebra identities (idempotent, null, identity, complement, absorption, De Morgan, etc.) enable stepwise elimination of redundant literals and terms. These transformations map to fewer or simpler gates when implemented. Step-by-Step Solution: Step 1: Write the expression in SOP or POS form.Step 2: Apply identities to merge terms and drop redundancies.Step 3: Optionally validate with Karnaugh maps or truth tables. Verification / Alternative check: Compare outputs for all input combinations before and after reduction; they match, confirming equivalence. Why Other Options Are Wrong: Incorrect: Conflicts with standard practice across textbooks and tools.Only possible with Karnaugh maps: K-maps are one method; Boolean algebra works directly as well.Only possible with computer synthesis: Manual algebraic reduction predates automated tools. Common Pitfalls: Stopping early and leaving near-redundant terms.Mixing arithmetic intuition with Boolean rules. Final Answer: Correct

Question 11

In Boolean algebra, evaluate the identity A + 1 = 1 for any Boolean variable A (domination law).

Accepted Answer

Introduction / Context: In Boolean algebra, identities describe how logical variables combine under the OR (+) and AND (*) operations. The expression A + 1 = 1 is known as a domination (or annihilator) law for OR, asserting that ORing any Boolean variable A with logic 1 yields logic 1 regardless of the value of A. This question checks whether you recognize and correctly apply this fundamental identity used in simplifying logic expressions and digital circuits. Given Data / Assumptions: A is a Boolean variable, so A can be 0 or 1. Symbol + denotes Boolean OR. We assume standard Boolean algebra axioms used in digital logic design. Concept / Approach: The domination law for OR states: X + 1 = 1 for any Boolean X. Intuitively, if at least one input to an OR is 1, the output is 1. Therefore, the presence of 1 renders the other input irrelevant to the final value. Recognizing this lets you immediately collapse expressions of the form A + 1 + B to 1, which is valuable in gate-level simplification. Step-by-Step Solution: Consider A = 0: 0 + 1 = 1.Consider A = 1: 1 + 1 = 1 (since 1 OR 1 = 1).Since both cases evaluate to 1, A + 1 = 1 holds for all A.Therefore, the statement is correct for every Boolean assignment. Verification / Alternative check: Truth table verification across A in {0, 1} confirms the identity. Alternatively, by duality, the corresponding AND annihilator is A * 0 = 0. Both are foundational and appear in standard identity sets used for Karnaugh maps and algebraic reductions. Why Other Options Are Wrong: Incorrect: Contradicted by the truth table; there is no assignment making A + 1 equal 0. Ambiguous as stated: The operators and domain are unambiguous in Boolean algebra. Cannot be determined: No additional data is required; identities are universal. Common Pitfalls: Confusing arithmetic addition with Boolean OR; in Boolean algebra, the symbol + represents OR, not numeric addition. Also, do not assume A + 1 = A + 1 (unchanged); the law collapses it to 1 immediately. Final Answer: Correct

Question 12

In canonical forms, assess the statement: “The product-of-sums (POS) is basically the ORing of ANDed terms.”

Accepted Answer

Introduction / Context: Canonical Boolean forms are widely used in analysis, simplification, and hardware realization. Two key canonical forms are sum-of-products (SOP) and product-of-sums (POS). This question probes whether you can distinguish their structural definitions and not confuse the operator hierarchy involved in each form. Given Data / Assumptions: SOP consists of ORing multiple product (AND) terms. POS consists of ANDing multiple sum (OR) terms. Operators: + is OR, * is AND (often omitted in writing). Concept / Approach: By definition, POS is the AND of sums (OR terms). The statement claims it is the OR of ANDed terms, which actually describes SOP. Therefore, the statement swaps the roles of SOP and POS and is incorrect. Step-by-Step Solution: SOP example: F = AB + A~B + ~AC (OR of AND terms).POS example: F = (A + B)(~A + C)*(B + ~C) (AND of OR terms).Compare with the statement: “ORing of ANDed terms” = SOP, not POS.Hence, the statement about POS is incorrect. Verification / Alternative check: Standard textbooks and logic design courses define POS as a conjunction (product) of maxterms (which are sums), while SOP is a disjunction (sum) of minterms (which are products). This cross-check reinforces the conclusion. Why Other Options Are Wrong: Correct: Conflicts with canonical definitions. Ambiguous as stated: The phrasing is explicit; no ambiguity about operators. Cannot be determined: Definitions are fixed; no extra data needed. Common Pitfalls: Memorizing acronyms without internalizing structure. A quick mnemonic: SOP = Sum (OR) of Products (AND); POS = Product (AND) of Sums (OR). Final Answer: Incorrect

Question 13

Evaluate the claim: “Five-variable Karnaugh maps are impossible.” Consider standard K-map construction practices.

Accepted Answer

Introduction / Context: Karnaugh maps (K-maps) provide a visual method for simplifying Boolean expressions by grouping adjacent 1s (or 0s) according to Gray-coded adjacency. While 4-variable K-maps are commonly shown in textbooks, higher-variable maps also exist and are used in practice for instructional and limited design tasks. Given Data / Assumptions: We consider the existence and usability of a 5-variable map. Adjacency follows Gray code ordering. Goal is simplification by grouping power-of-two cells. Concept / Approach: Five-variable K-maps are typically realized as two linked 4-variable maps (or a 32-cell layout), where the fifth variable toggles between the maps. Groups can wrap across the two maps to preserve adjacency, enabling simplification just as with 3- or 4-variable cases, albeit with more care. Step-by-Step Solution: Represent variables as A, B, C, D in a 4x4 grid using Gray code.Introduce the fifth variable E by drawing two 4-variable maps: one for E = 0 and one for E = 1.Mark minterms in the appropriate map; consider adjacency both within each map and across the pair.Form groups of 1, 2, 4, 8, or 16 cells that respect adjacency rules, including wrap-around and inter-map adjacency. Verification / Alternative check: Numerous educational resources and CAD tools demonstrate 5-variable K-map groupings. Although 6-variable K-maps are also possible (often as four linked 4-variable maps), the method becomes unwieldy, motivating algorithmic minimization (e.g., Quine–McCluskey) for larger problems. Why Other Options Are Wrong: Correct: Contradicts standard construction using two linked maps. Ambiguous as stated: The statement is categorical; there is no ambiguity in “impossible.” Cannot be determined: The existence is factual; no extra data is required. Common Pitfalls: Assuming K-maps end at 4 variables. Usability decreases with more variables, but possibility remains. Do not confuse “impractical” with “impossible.” Final Answer: Incorrect

Question 14

Tool flow check: A VHDL editor bundled with a schematic editor can generate a .vhd file that the simulator can use to test a logic design. Evaluate this statement.

Accepted Answer

Introduction / Context: Digital design toolchains commonly integrate schematic capture, hardware description language (HDL) editing (e.g., VHDL), synthesis, and simulation. Understanding how design artifacts move between these tools helps validate whether a given workflow statement is reasonable. Given Data / Assumptions: The environment includes a schematic editor and a VHDL editor. The editor emits .vhd (VHDL source) files. A simulator accepts VHDL files to verify design behavior. Concept / Approach: VHDL is a standard HDL used to describe digital logic structurally or behaviorally. Many vendor and third-party suites allow editing VHDL, then invoking a simulator to run testbenches and observe waveforms. Schematic blocks can often be netlisted to HDL or combined with HDL blocks for co-simulation. Therefore, generating .vhd and consuming it in a simulator is a normal, supported flow. Step-by-Step Solution: Use the VHDL editor to create or export entity/architecture code as a .vhd file.Provide or link a testbench that instantiates the DUT (device under test).Load both files into a VHDL simulator.Run simulations, inspect signals, and iterate on the design. Verification / Alternative check: Typical flows in popular EDA tools (vendor or academic) follow this pattern. Whether code originates from schematic-to-HDL netlisting or hand-written VHDL, the simulator operates on .vhd and testbench sources. Why Other Options Are Wrong: Incorrect: It contradicts widely used, standard workflows. Ambiguous as stated: The file extension and simulator usage are explicit. Cannot be determined: No extra data is needed; it describes a common capability. Common Pitfalls: Confusing synthesis vs simulation artifacts. Netlists post-synthesis may be different, but simulators routinely accept behavioral or structural VHDL for functional testing. Final Answer: Correct

Question 15

Clarify the acronym: In digital logic, does SOP stand for “sum-of-powers” or something else?

Accepted Answer

Introduction / Context: Acronyms in digital logic often encode structural meanings. SOP is foundational in Boolean simplification and hardware realization. Misinterpreting the acronym can cause confusion about canonical forms and impede correct minimization strategies. Given Data / Assumptions: We are evaluating whether SOP expands to “sum-of-powers.” Standard operators: + is OR (sum), * is AND (product). We assume conventional terminology from digital logic curricula. Concept / Approach: SOP stands for “sum-of-products,” meaning an OR (sum) of product (AND) terms, typically minterms or implicants. “Sum-of-powers” is not a Boolean algebra term; it appears to be a mishearing of “products.” Step-by-Step Solution: Identify SOP’s structure: ORing together multiple ANDed literals/variables.Example: F = AB + ~AC + B*C.Compare with the claim “sum-of-powers”: not a recognized construct in Boolean algebra.Therefore, the statement “SOP stands for sum-of-powers” is invalid. Verification / Alternative check: In canonical SOP, each product term corresponds to a minterm (all variables present, possibly complemented). Textbook definitions and K-map procedures rely on “products,” not “powers.” Why Other Options Are Wrong: Correct: Would endorse an incorrect expansion. Ambiguous as stated: The acronym expansion is precise, not ambiguous. Cannot be determined: Standardized terminology provides the answer. Common Pitfalls: Hearing or reading mistakes between “products” and “powers.” Remember: products arise from ANDing literals; sums arise from ORing terms. Final Answer: Incorrect

Question 16

Compare representations: A Karnaugh map (K-map) is similar to a truth table because it enumerates all input combinations and the corresponding output values. Evaluate this statement.

Accepted Answer

Introduction / Context: Karnaugh maps and truth tables are two complementary views of the same function: a truth table lists outputs for all possible input combinations, while a K-map lays out those combinations spatially to make adjacency and grouping visually apparent for simplification. This question checks your understanding of their relationship. Given Data / Assumptions: Both tools describe the same Boolean function. Gray code ordering is used in K-maps to enforce single-bit adjacency. The purpose is simplification by identification of prime implicants. Concept / Approach: The K-map is a reorganized truth table where each cell corresponds to a minterm (for SOP) or maxterm (for POS). The spatial arrangement helps you see powers-of-two groupings that translate into simplified product or sum terms, reducing literal counts and gate complexity. Step-by-Step Solution: Start with a truth table: enumerate all 2^n input combinations.Map each combination to a K-map cell via Gray-coded rows/columns.Place 1s (or 0s) into cells according to the function's output.Group adjacent 1s (or 0s) to derive simplified expressions. Verification / Alternative check: Translating a function to both a truth table and a K-map yields identical functional information. The only difference is layout and the resulting ease of visual minimization. Why Other Options Are Wrong: Incorrect: Denies the shared enumeration of input states and outputs. Ambiguous as stated: The mapping between rows and cells is well-defined. Cannot be determined: No additional data is required; definitionally true. Common Pitfalls: Thinking K-maps introduce new semantics beyond the truth table. They do not; they simply reorganize data to make adjacency-based simplification obvious. Final Answer: Correct

Question 17

Definition check: In digital logic, a “variable” denotes a symbol representing a logical quantity that can assume the value 1 or 0. Evaluate this statement.

Accepted Answer

Introduction / Context: Precise terminology is essential in Boolean algebra and digital systems. A variable is a placeholder for a logical value that can change according to inputs, states, or time. Understanding this definition underpins work with truth tables, K-maps, and HDL descriptions. Given Data / Assumptions: The logical domain is binary: {0, 1}. Variables may be complemented (~A) or uncomplemented (A). Expressions are composed using OR (+), AND (*), and NOT (~). Concept / Approach: A Boolean variable represents a logical quantity that can be either 0 (false/LOW) or 1 (true/HIGH). This is distinct from multi-valued logic systems (e.g., ternary or 4-state simulation), which extend beyond basic Boolean algebra typically taught in introductory courses. Step-by-Step Solution: Interpret “variable” as a symbol like A, B, or C.Allowable values in Boolean algebra: 0 or 1.Use in expressions and truth tables assumes binary evaluation.Therefore, the statement correctly defines a Boolean variable. Verification / Alternative check: Truth tables treat each variable as taking 0 or 1, enumerating 2^n combinations for n variables. HDL signals in simple combinational contexts also behave as binary variables under idealized models. Why Other Options Are Wrong: Incorrect: Conflicts with the standard definition in Boolean logic. Ambiguous as stated: The statement specifies values clearly (0 or 1). Cannot be determined: No further information is needed; it is definitional. Common Pitfalls: Conflating Boolean variables with arithmetic variables that can take many integer values. In logic, the domain is strictly binary unless otherwise noted (e.g., X/Z states in simulation models). Final Answer: Correct

Question 18

Repair and evaluate the idempotent identity: In Boolean algebra, A + A = A (idempotent law for OR). Is this identity valid for all Boolean A?

Accepted Answer

Introduction / Context: The original stem was incomplete. Applying the Recovery-First Policy, we minimally repair it to the well-known idempotent identity A + A = A, which expresses that ORing a Boolean variable with itself yields the same variable. Idempotent laws (for both OR and AND) are among the most frequently used rules in algebraic simplification and K-map reasoning. Given Data / Assumptions: A is Boolean (0 or 1). + denotes Boolean OR. Goal: validate A + A = A for all A. Concept / Approach: An operation is idempotent if applying it multiple times does not change the result beyond the first application. For OR, combining A with itself does not introduce any new information beyond A, so the result equals A. There is a dual idempotent law for AND: A * A = A. Step-by-Step Solution: Case A = 0: 0 + 0 = 0, which equals A.Case A = 1: 1 + 1 = 1, which equals A.Since both cases match A, the identity holds universally.Thus, A + A simplifies to A in any Boolean expression. Verification / Alternative check: Truth table enumeration confirms the law instantly. In set-theoretic interpretation (union), X ∪ X = X mirrors the same idempotency, offering further intuition for the identity's correctness. Why Other Options Are Wrong: Incorrect: Contradicted by both truth table and set analogy. Ambiguous as stated: After repair, the identity is precise and standard. Cannot be determined: No further data is needed to validate an algebraic law. Common Pitfalls: Confusing with absorption (A + A*B = A) or domination (A + 1 = 1). While related, idempotency is distinct and should not be interchanged with those identities. Final Answer: Correct

Question 19

Role of Boolean algebra: Assess the statement “Boolean algebra simplifies logic circuits.”

Accepted Answer

Introduction / Context: Boolean algebra provides the theoretical foundation for reasoning about digital circuits. It enables transformation of logic expressions to equivalent but simpler forms, which in turn reduces hardware cost, power, and delay. This question checks recognition of Boolean algebra’s core purpose in design flows. Given Data / Assumptions: We are evaluating the general role of Boolean algebra in circuit simplification. No specific function is given; the claim is about capability and intent. “Simplifies” covers fewer gates, fewer inputs per gate, or reduced literal count. Concept / Approach: Using identities (e.g., De Morgan, idempotent, absorption, consensus), designers rewrite expressions into minimal or near-minimal forms. Karnaugh maps and algorithmic methods (Quine–McCluskey, heuristics) are systematic applications of Boolean principles to achieve simplification and reduction of implementation complexity. Step-by-Step Solution: Start from a given expression derived from a truth table.Apply identities to merge terms and eliminate redundancies.Verify equivalence by truth table or algebraic reasoning.Map the simplified expression to a gate-level implementation with fewer resources. Verification / Alternative check: Practical gains include fewer integrated circuits, simpler routing, and improved timing. Even when exact minima are hard to find, Boolean techniques reliably reduce complexity relative to a naive implementation. Why Other Options Are Wrong: Incorrect: Denies a fundamental and well-documented outcome of Boolean manipulation. Ambiguous as stated: The role is widely defined and accepted. Cannot be determined: The statement refers to a general capability, which is established. Common Pitfalls: Equating “simplify” solely with minimal gate count. In practice, designers balance multiple objectives, but Boolean simplification remains the first step toward efficiency. Final Answer: Correct

Question 20

Device choice: If a design requires a NAND gate, an AND function, and an inverter, would selecting a 7400 IC (quad 2-input NAND) be appropriate?

Accepted Answer

Introduction / Context: The 7400 is a classic TTL IC containing four independent 2-input NAND gates. Knowing how to realize multiple logical functions using only NANDs is a standard skill because NAND is functionally complete: you can build any combinational logic, including NOT and AND, from NANDs alone. Given Data / Assumptions: Available device: 7400 IC with four 2-input NAND gates. Required functions: one NAND, one AND, and one inverter (NOT). Assume unused gates are available on the same IC for composition. Concept / Approach: A NAND becomes an inverter by tying its inputs together (Y = NAND(A, A) = ~A). An AND function can be built by NAND followed by inversion: AND(A, B) = ~(~(A * B)) = invert(NAND(A, B)). Thus, a single 7400 provides enough gates to realize all three required behaviors simultaneously (subject to pin budgeting). Step-by-Step Solution: Use Gate 1 of the 7400 directly as NAND(A, B).Use Gate 2 as an inverter: tie its two inputs together to form ~C or to invert NAND output.Implement AND(A, B) by feeding NAND(A, B) into the inverter from Gate 2.Use Gate 3 as a separate inverter if you also need to invert an independent signal. Verification / Alternative check: Truth tables confirm that NAND with tied inputs realizes NOT, and NOT(NAND) realizes AND. The 7400's four gates typically provide enough resources for these combinations in small designs. Why Other Options Are Wrong: Incorrect: Overlooks NAND functional completeness and common implementation patterns. Ambiguous as stated: The device and functions are clearly specified. Cannot be determined: No additional data is needed beyond basic gate composition rules. Common Pitfalls: Forgetting to budget gates: although one 7400 usually suffices, ensure you count how many inverters and NANDs your complete design needs to avoid running out of gates. Final Answer: Correct

Boolean Algebra and Logic Simplification Questions