Truth Tables, Universal Gates, and De Morgan’s Laws Quiz

For n Boolean variables, each variable can be 0 or 1, leading to 2^n possible combinations of inputs. The truth table must include all these combinations to fully define the function, so it has 2^n rows. Fixed row counts like 2, n, or n^2 are incorrect for variable n.

A 3-variable function explicitly has three Boolean variables, such as A, B, and C. The number of rows in its truth table is 2^3 = 8, but the variables themselves are three. Options like 2, 6, or 8 refer to other aspects but not the variable count.

A 3-input AND gate outputs 1 only if all inputs A, B, and C are 1. The Boolean expression for this is A · B · C, denoting logical AND. A + B + C is for OR, (A · B · C)' is for NAND, and A ⊕ B ⊕ C is for XOR.

The OR gate computes X + Y, and this result is ANDed with Z, giving (X + Y) · Z. This order of operations is key; other expressions like X + (Y · Z) or X · Y + Z would require different gate arrangements.

When both inputs of an AND gate are connected to the same signal A, the output is A because A AND A equals A. This is derived from Boolean algebra: A · A = A. It is not inversion, constant 1, or constant 0.

NAND and NOR gates are universal, meaning that by using only NAND gates or only NOR gates, you can construct any other logic gate, such as AND, OR, NOT, etc. This is not true for AND-OR, XOR-XNOR, or NOT-AND pairs alone.

When both inputs of a NAND gate are tied to the same signal A, the output is (A · A)' = A', which is NOT A. This effectively creates a NOT gate. Connecting one input to 1 gives A', but it requires a constant 1, while tying both inputs is self-sufficient.

When both inputs of a NOR gate are connected to the same signal A, the output is (A + A)' = A', which is NOT A. This forms a NOT gate. Connecting one input to 0 also works, but tying both inputs is a common method without external constants.

NOR gates are universal gates, meaning that using only NOR gates, you can build any combinational logic circuit, including AND, OR, NOT, etc. NAND gates are also universal, but they aren't listed among the options. AND, XOR, or OR gates alone cannot construct all possible logic functions, making them non-universal gates.

De Morgan's Law states that (A · B)' = A' + B'. This means a NAND gate is equivalent to an OR gate with inverted inputs. This law allows the transformation between AND-OR and NAND-NOR implementations in logic design.

De Morgan's Law states that (A + B)' = A' · B'. This means a NOR gate is equivalent to an AND gate with inverted inputs. This is used to simplify circuits and convert between gate types.

To make an OR gate from NOR gates, first use a NOR gate with inputs A and B, which outputs (A + B)'. Then, invert this output using another NOR gate with both inputs tied to (A + B)', which gives ((A + B)')' = A + B. Thus, two NOR gates are used: one as NOR and one as an inverter.

A universal gate must be able to construct all other logic gates (AND, OR, NOT, etc.). While an XOR gate can function as a NOT gate when one input is held constant at 1, it cannot be used to create an AND gate or OR gate using only XOR gates. Without the ability to construct all basic logic functions, XOR is not considered a universal gate. NAND and NOR gates, by contrast, are universal because any logic function can be built using only NAND gates or only NOR gates.

For a gate with 4 inputs, each input can be 0 or 1, so the total combinations are 2^4 = 16. This is derived from the formula 2^n for n inputs. Options like 4, 8, or 32 are incorrect for n=4.

The XOR gate outputs A ⊕ B (often denoted by ⊕ or ⊕), and the NOT gate inverts it to (A ⊕ B)'. This is the XNOR operation, which outputs 1 when inputs are the same. Thus, the final expression is (A ⊕ B)'.