Question

For Boolean identities match the pair: (1) \(\underline{\underline{A}}\) (P) \(\underline{\mathrm{A}}+\underline{\mathrm{B}}\) (2) \(\underline{\mathrm{A}+\mathrm{B}}\) (Q) A \cdot B (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) (R) \(\underline{A} \cdot \underline{B}\) (4) \(\mathrm{A} \cdot(\underline{\mathrm{A}}+\mathrm{B})\) (S) \(\mathrm{A}\) (A) \((1)-(\mathrm{S}),(2)-(\mathrm{P}),(3)-(\mathrm{Q}),(4)-(\mathrm{R})\) (B) \((1)-(\mathrm{S}),(2)-(\mathrm{R}),(3)-(\mathrm{Q}),(4)-(\mathrm{P})\) (C) (1) - (S), (2) - (Q), (3) - (P), (4) - (R) (D) \((1)-(\mathrm{S}),(2)-(\mathrm{R}),(3)-(\mathrm{P}),(4)-(\mathrm{Q})\)

Short answer

The short answer is: (D) (1) - (S), (2) - (R), (3) - (P), (4) - (Q)

Step-by-step solution

1) Simplifying the given Boolean identities expressions

Let's start by simplifying or rearranging the expressions: (1) \(\underline{\underline{A}}\) : The complement of complement A is simply A. (2) \(\underline{\mathrm{A}+\mathrm{B}}\) : This is the complement of the sum of A and B. (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) : This is the complement of the product of A and B. (4) \(\mathrm{A} \cdot(\underline{\mathrm{A}}+\mathrm{B})\) : Simplifying this expression using the distributive law of Boolean algebra, we have \(A \cdot \underline{A} + A \cdot B\). And since \(A\cdot\underline{A}=0\), it becomes \(0 + A \cdot B\), which is simply \(A \cdot B\). Now let's move to the equivalent expressions.

2) Identify the equivalent expressions

For each original expression (1, 2, 3, and 4), we want to associate it with the equivalent expression (P, Q, R, and S): (P) \(\underline{\mathrm{A}}+\underline{\mathrm{B}}\) : Using De Morgan's law, this simplifies to \(\underline{A \cdot B}\). (Q) A \(\cdot\) B : This is simply the product of A and B. (R) \(\underline{A} \cdot \underline{B}\) : Using De Morgan's law, this simplifies to \(\underline{A + B}\). (S) A : This is simply A.

3) Match the pairs

Now that we have simplified and/or identified the equivalent expressions for each Boolean identity, we can match the pairs as follows: (1) A -> (S) A (2) \(\underline{\mathrm{A}+\mathrm{B}}\) -> (R) \(\underline{A \cdot B}\) (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) -> (P) \(\underline{A} + \underline{B}\) (4) \(A \cdot B\) -> (Q) A \(\cdot\) B So the correct choice is given by option (D): (D) (1) - (S), (2) - (R), (3) - (P), (4) - (Q)