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E. Assertion and Reasoning Based Questions.
The following questions consist of two statements – Assertion (A) and Reason (R). Answer these questions by selecting the
appropriate option given below:
a. Both A and R are true and R is the correct explanation of A.
b. Both A and R are true but R is not the correct explanation of A.
c. A is true but R is false.
d. A is false but R is true.
1. Assertion (A): A compound proposition is called a tautology if it is true for all possible truth values of its propositional variables.
Reason (R): The proposition is evaluated as 0 or false in the last column of the truth table.
Ans. c. A is true, but R is false.
2. Assertion (A): In De Morgan’s law, (a ∧ b)' = a' ∨ b' and (a ∨ b)' = a' ∧ b'.
Reason (R): De Morgan’s law helps simplify expressions by swapping AND and OR operations with their complements.
Ans. a. Both A and R are true and R is the correct explanation of A.
3. Assertion (A): For the conditional a → b, the converse a → b and the contrapositive a' → b' are logically equivalent.
Reason (R): The contrapositive of a conditional is obtained by taking the converse of the inverse of the conditional.
Ans. d. A is false, but R is true.
F. Case study-based questions.
A group of engineers is working on designing a digital system to perform binary addition. They need to create an efficient and
effective way to add binary numbers. After conducting some research, they learn that adders are the building blocks of such systems
and they focus on two types of adders: Half Adder and Full Adder. These circuits are essential for handling binary addition and carry
operations, which are crucial in computer systems.
Half Adder: It is a simple combinational circuit used to add two single-bit binary numbers. The output consists of two bits: the sum
and the carry. The Boolean expressions for the sum and carry are:
• Sum = A'B + AB'
• Carry = AB
Full Adder: It is a more complex circuit that adds three binary digits: two input bits and the carry bit from a previous addition. It
outputs two bits: the sum and the carry-out. The Full Adder can be thought of as a combination of two Half Adders and an additional
OR gate to handle the carry-out.
Based on the given case, answer the following questions:
1. What are the two types of adders used in digital systems?
Ans. Half Adder and Full Adder.
2. What are the inputs and outputs of a Half Adder?
Ans. The inputs of a Half Adder are two single-bit binary numbers, A and B. The outputs are the sum and carry bits (Sum and Carry).
3. Write the Boolean expression for the Sum and Carry in a Half Adder.
Ans. • Sum = A'B + AB'
• Carry = AB
4. What logic gates are used in the Half Adder circuit?
Ans. The Half Adder uses an XOR gate for the sum and an AND gate for the carry.

Unsolved Questions

A. Tick ( ) the correct option.

1. The propositional operator → represents ………………… .
a. conjunction b. implication
c. disjunction d. negation

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