Page 51 - Computer science 868 Class 12
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2. Consider the following propositions
P = “It is raining heavily.”
Q = “There is a prediction for cyclone.”
R = “There will be damage of life and property.”
Now, write the following statements in symbolic form:
a. If there is a prediction for cyclone, then it is raining heavily and there will be damage to life and property.
b. It is raining heavily and there is a prediction for cyclone.
c. There is a prediction for cyclone if and only if it is raining heavily.
d. It is not raining heavily or there is no prediction of cyclone.
e. If there is a prediction for cyclone then there will be damage to life and property.
3. Consider the following propositions
P = “It is raining heavily.”
Q = “There is a prediction for cyclone.”
R = “There will be damage to life and property.”
Now, express the following expressions in words:
a. ∼P ∨ ∼Q
b. ∼P ∨ (∼Q ∧ ∼R)
c. ∼P ↔ ∼Q
d. ∼(P ∧ Q ∧ R)
e. ∼Q → ∼R
4. Write the converse, inverse and contrapositive for the following propositions:
i. If you work hard, then you will crack IIT.
ii. If the flower has a beautiful smell, then it is a rose.
iii. If I am invited to a party, then I will wear my red gown.
iv. If India wins the World Cup cricket match, then we will celebrate.
v. If a number is equal to its reverse, then it is a palindrome.
5. Construct the truth tables and check if the following propositions are tautology, contradiction or contingency:
a. (a ↔ b) → (a ∧ b)
b. b → (∼a → b) ↔ a
c. (a ∧ b) → (∼a ∧ ∼b)
d. [(a → b) → (b → c)] → a
e. (a ∧ 0) → (∼a ∧ b)
f. a ∧ (a → b) → b
g. (a ↔ b) ∧ (∼a ↔ ∼b)
h. (a ∧ b) → (∼a ∧ b) → (∼b ∧ a)
i. ∼(a ∨ b) ∧ (a ∨ b)
j. ∼(a ∧ b ∧ c) ∨ (a ∧ b ∧ c)
6. Complete the truth table given below.
P Q P → Q (P → Q) → P' ∼ ((P → Q) → ∼P)
7. Minimise the following Boolean expressions using the laws of Boolean Algebra:
a. X.Y + X'.Z + Y.Z = X.Y + X'.Z
b. A.(B + C.(A+B.C)')'
c. (X'+Y'+Z).(X'+Y'+Z') + (X+Y+Z).(X+Y+Z')
d. (X+X'.Y).(X'+Y')
e. A.B.C + A.B.C' + A'.B.C + A.B'.C
f. X'.Y + X'.Y' + (Z.Z)' + Z
g. A.B + A.B' + A'.B'
h. (X' + (X.Y)' + Z)'
i. A.(A'+B).C.(A+B) = A.B.C [ISC 2018]
j. F = (x'+z) + [(y'+z).(x'+y)]' [ISC 2017]
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Boolean Algebra 49

