Boolean Expressions and Logic Circuit Design Quiz

A Boolean function takes one or more Boolean variables as inputs and maps them to an output value in the set {0, 1}. This binary output defines the function's behavior, such as in truth tables. It does not map variables to constants arbitrarily, constants to variables, or operators to variables.

The truth table shows output 1 when at least one input is 1, which is characteristic of an OR gate. An AND gate outputs 1 only for (1,1), XOR outputs 1 only when inputs differ, and NAND outputs 0 only for (1,1), so none match except OR.

For n Boolean variables, the truth table must list all possible combinations of inputs. Each variable has 2 states (0 or 1), so the total number of rows is 2^n. This ensures all input scenarios are covered, unlike fixed row counts like 2, n, or n^2.

This truth table outputs 1 when the inputs are different, which is the definition of an XOR gate. XNOR outputs 1 when inputs are the same, AND outputs 1 only for (1,1), and OR outputs 1 for any input except (0,0), so only XOR fits.

The expression A · B + C means first compute A AND B, then OR the result with C. So, A and B are inputs to an AND gate, and the output of that AND gate is fed along with C into an OR gate. This order of operations is critical in logic circuit design.

The expression (A + B) · C means first compute A OR B, then AND the result with C. Thus, A and B are inputs to an OR gate, and the output of that OR gate is fed along with C into an AND gate. This implements the logical AND of the OR result with C.

The expression A' + B requires first inverting A to get A', then ORing that with B. So, A is connected to a NOT gate, and the output of the NOT gate (A') and B are connected to an OR gate. Other configurations would yield different expressions, like (A + B)' or A' · B.

The expression (A · B)' is the negation of A AND B, which is exactly the function of a NAND gate. A NAND gate outputs 0 only when both inputs are 1, and 1 otherwise, matching (A · B)'. Other gates like AND, NOR, or OR do not directly implement this.

To implement A' · B', first invert A and B using two NOT gates to get A' and B'. Then, feed A' and B' into an AND gate. This produces the output A' · B'. Using OR gates or fewer NOT gates would not yield the correct expression.

The expression A + B + C' requires inverting C to get C', then ORing A, B, and C'. So, one NOT gate is used for C, and a 3-input OR gate combines A, B, and C'. Using AND gates or multiple NOT gates would not achieve the OR operation.

XOR is often denoted by ⊗ or ⊕. Inverting the XOR result gives XNOR, which is expressed as (A ⊕ B)'. This output is 1 when inputs are the same. (A + B)' is NOR, (A · B)' is NAND, and A ⊕ B is XOR without inversion.

The expression A · (B + C) means first compute B OR C using an OR gate, then AND that result with A using an AND gate. Thus, one OR gate and one AND gate are needed. Other combinations would not correctly implement the expression.

The XOR operation can be implemented as (A AND NOT B) OR (NOT A AND B), which is expressed as (A · B') + (A' · B). This requires two NOT gates, two AND gates, and one OR gate. Option b represents XNOR, while options a and d are insufficient to implement XOR.

The AND gate computes A · B, and the NOT gate inverts it, resulting in (A · B)'. This is the NAND operation. Other options like A' · B' would require NOT gates before AND, and (A + B)' would require an OR gate followed by NOT.

The NOT gate inverts A to A', and then A' is ORed with B, giving A' + B. This is the direct implementation. Option a is A OR B, c is A OR NOT B, and d is the negation of A' OR B, which is not correct.