In Boolean algebra, the value of the expression A added to the complement of A is

The correct answer is B, OR. Read the table row by row. The output is 0 only when both inputs are 0, and it is 1 in the other three rows, including the row where both inputs are 1. That is exactly the rule of an OR gate, which gives 1 when at least one input is 1 and behaves like two switches wired in parallel. A is wrong because a NOT gate has only one input and so cannot have a table with two input columns at all. C is wrong because an XOR gate responds only to a difference between its inputs, so its last row, with both inputs 1, would give 0 and not 1; this is the one row that separates OR from XOR and the reason the distractor is offered. D is wrong because an AND gate gives 1 only in the last row and 0 in the first three, which is the mirror image of the table shown.

The correct answer is B, NAND and NOR. Each of these gates alone is enough to build every other gate and so every logic circuit: a NAND with its two inputs tied together acts as a NOT, two NANDs in sequence give an AND, and a suitable arrangement of three gives an OR, and the same can be done entirely with NOR gates. That is why chip makers sell packages containing only one gate type. A is wrong because AND and OR cannot produce a complement by themselves; without a NOT they can never invert a signal. C is wrong because XOR and XNOR are themselves derived gates, built from the basic three, and neither can generate the full set on its own. D is wrong because NOT with AND can indeed build everything, but the pair is not given the name universal gates; the term is reserved for the two single gates that suffice by themselves.

The correct answer is B, the sum of the complements. De Morgan's second theorem states that NOT of A AND B equals NOT A OR NOT B. The working rule is to break the bar and change the sign, so a dot under a complement becomes a plus once the complement is distributed over the variables. A is wrong because the product of the complements is the result of the first theorem, which applies to the complement of a sum, not of a product; swapping the two theorems is the standard error in this question. C is wrong because complementing an expression must change it unless the expression is a constant, and the product of A and B is not its own complement. D is wrong because the value depends on the inputs: when A is 1 and B is 1 the expression is 0, so it cannot always be 1. Both theorems together make NAND-only and NOR-only design possible.

The correct answer is C, when the two inputs are different. An exclusive OR gate is a difference detector: it gives 1 for the combinations 0 and 1 or 1 and 0, and gives 0 when the inputs agree. It is sometimes read as either but not both. A is wrong because two inputs of 1 give an output of 0 in an XOR gate; that row is exactly what distinguishes it from an ordinary OR gate, which would give 1. B is wrong because two inputs of 0 also agree, so the output is again 0. D is wrong because an output of 1 for identical inputs describes the XNOR gate, the complement of XOR, which works as an equality detector and is used to compare two binary numbers bit by bit. In a half adder, the XOR gate supplies the sum bit while the AND gate supplies the carry.

The correct answer is D, a variable in its complemented or uncomplemented form. In Boolean algebra a literal is any single appearance of a variable in an expression, whether it appears plain or with a bar over it. The count of literals is used to measure how costly an expression is, because each literal becomes one input line to a gate, so simplification is judged by how many literals it removes. A is wrong because restricting the term to the plain form would leave no name for the complemented appearance, which is equally a literal. C is wrong for the mirror reason: the complemented form is not the only kind. B is wrong because, read as it stands, it suggests a variable taken together with its complement, which describes a pair rather than the single appearance that a literal is. Remember that a term such as A AND NOT B contains two literals and two variables.