HSC Commerce (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Write the converse, inverse, and contrapositive of the following statement.

If he studies, then he will go to college.

With proper justification, state the negation of the following.

(p → q) ∨ (p → r)

With proper justification, state the negation of the following.

(p ↔ q) v (~ q → ~ r)

With proper justification, state the negation of the following.

(p → q) ∧ r

Construct the truth table for the following statement pattern.

(p ∧ ~ q) ↔ (q → p)

Construct the truth table for the following statement pattern.

(~p ∨ q) ∧ (~p ∧ ~q)

Construct the truth table for the following statement pattern.

(p ∧ r) → (p ∨ ~q)

Construct the truth table for the following statement pattern.

(p ∨ r) → ~(q ∧ r)

Construct the truth table for the following statement pattern.

(p ∨ ~q) → (r ∧ p)

What is tautology? What is contradiction?
Show that the negation of a tautology is a contradiction and the negation of a contradiction is a tautology.

Determine whether the following statement pattern is a tautology, contradiction, or contingency.

[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]

Determine whether the following statement pattern is a tautology, contradiction, or contingency.

[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)

Determine whether the following statement pattern is a tautology, contradiction, or contingency.

[~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]

Determine whether the following statement pattern is a tautology, contradiction, or contingency.

[~(p ∧ q) → p] ↔ [(~p) ∧ (~q)]

Determine whether the following statement pattern is a tautology, contradiction, or contingency.

[p → (~q ∨ r)] ↔ ~[p → (q → r)]

Using the truth table, prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Using the truth table, prove the following logical equivalence.

[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

Using the truth table, prove the following logical equivalence.

p ∧ (~p ∨ q) ≡ p ∧ q

Using the truth table, prove the following logical equivalence.

p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

Using the truth table, prove the following logical equivalence.

~p ∧ q ≡ [(p ∨ q)] ∧ ~p