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4. Verify the following proposition with the help of a truth table:
(P ∧ Q) ∨ (P ∧ ∼Q) = P [ISC 2018]
Ans. P Q P ∧ Q P ∧ ∼Q (P ∧ Q) ∨ (P ∧ ∼Q)
0 0 0 0 0
0 1 0 0 0
1 0 0 1 1
1 1 1 0 1
We derive that column 1 and column 5 have the same combinations of 0 and 1. Hence proved.
5. Using the truth table, state whether the following proposition is a tautology, contingency or a contradiction:
∼(A ∧ B) ∨ (∼A → B) [ISC 2018]
Ans. A B ∼(A ∧ B) (∼A → B) ∼(A ∧ B) ∨ (∼A → B)
0 0 1 0 1
0 1 1 1 1
1 0 1 1 1
1 1 0 1 1
The final column has a value 1. Hence, it is a tautology.
6. Differentiate between half adder and full adder. Draw the logic circuit diagram for a full adder. [ISC 2018]
Ans. Half Adder Full Adder
Half adder performs the addition of 2 bits. Full adder performs the addition of 3 bits.
The logic circuit diagram of a full adder circuit is:
A Sum = A ⊕ B ⊕ C
B in
C in
A.B
A.B+B.C +C .A
in
in
B.C in
C .A
in
7. Verify using truth table F = (P → Q) ∨ (Q → P'). [ISC 2020]
Ans. P Q (P → Q) (Q → P') (P → Q) ∨ (Q → P')
0 0 1 1 1
0 1 1 1 1
1 0 0 1 1
1 1 1 0 1
The final column has a value 1. Hence, it is a tautology.
95
Propositional Logic, Hardware Implementation, Arithmetic Operations 95

