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State the dual of the following statement by applying the principle of duality. p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)] - Mathematics and Statistics

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प्रश्न

State the dual of the following statement by applying the principle of duality.

p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]

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उत्तर

p ∧ (q ∧ r) ≡ ~[(p ∨ q) ∧ (r ∧ s)]

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पाठ 1: Mathematical Logic - Miscellaneous Exercise 1 [पृष्ठ ३३]

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बालभारती Mathematics and Statistics 1 (Commerce) [English] 12 Standard HSC Maharashtra State Board
पाठ 1 Mathematical Logic
Miscellaneous Exercise 1 | Q 4.16 | पृष्ठ ३३

संबंधित प्रश्‍न

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

[(p→q) ∧ q]→p


Prove that the following statement pattern is equivalent :

(p ∨ q) →  r and (p → r) ∧ (q → r)


If   p : It is raining
     q : It is humid

Write the following statements in symbolic form:

(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid. 


Use the quantifiers to convert the following open sentence defined on N into true statement:
x2 ≥ 1


Write the negation of the following statement : 
If the lines are parallel then their slopes are equal.


Using the truth table prove the following logical equivalence.

p → (q → p) ≡ ∼ p → (p → q)


Using the truth table prove the following logical equivalence.

(p ∨ q) → r ≡ (p → r) ∧ (q → r)


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

(p ↔ q) ∧ (p → ∼ q)


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

[(p ∨ ∼q) ∨ (∼p ∧ q)] ∧ r


Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)


Prepare truth table for (p ˄ q) ˅ ~ r

(p ∧ q) ∨ ~ r


Fill in the blanks :

Inverse of statement pattern p ↔ q is given by –––––––––.


Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p


Prove that the following pair of statement pattern is equivalent.

~(p ∧ q) and ~p ∨ ~q


Write the dual of the following:

(p ∨ q) ∨ r


Write the dual of the following:

~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]


Write the dual of the following:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r


Write the dual of the following:

~(p ∧ q) ≡ ~ p ∨ ~ q


Write the dual statement of the following compound statement.

13 is prime number and India is a democratic country.


Write the negation of the following statement.

All the stars are shining if it is night.


Write the negation of the following statement.

∀ n ∈ N, n + 1 > 0


Write the negation of the following statement.

∃ n ∈ N, (n2 + 2) is odd number.


Using the rules of negation, write the negation of the following:

(~p ∧ q) ∧ (~q ∨ ~r)


Construct the truth table for the following statement pattern.

(p ∨ r) → ~(q ∧ r)


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) ∧ (q ∧ r)] ∨ (~q)


Using the truth table, prove the following logical equivalence.

~p ∧ q ≡ [(p ∨ q)] ∧ ~p


State the dual of the following statement by applying the principle of duality.

2 is even number or 9 is a perfect square.


Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”


Choose the correct alternative:

If p → q is an implication, then the implication ~q → ~p is called its


The contrapositive of p → ~ q is ______


Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.


Complete the truth table.

p q r q → r r → p (q → r) ˅ (r → p)
T T T T `square` T
T T F F `square` `square`
T F T T `square` T
T F F T `square` `square`
F T T `square` F T
F T F `square` T `square`
F F T `square` F T
F F F `square` T `square`

The given statement pattern is a `square`


The equivalent form of the statement ~(p → ~ q) is ______.


In the triangle PQR, `bar(PQ) = 2bara and bar(QR)` = `2 bar(b)` . The mid-point of PR is M. Find following vectors in terms of `bar(a) and bar(b)` .

  1. `bar(PR)`  
  2. `bar(PM)`
  3. `bar(QM)`

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