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Write the dual of the following (p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q) - Mathematics and Statistics

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प्रश्न

Write the dual of the following

(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)

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उत्तर

(p ˅ ∼q) ˄ (∼p ˅ q) ≡ (p ˄ q) ˅ ∼(p ˅ q) 

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अध्याय 1.1: Mathematical Logic - Q.4

संबंधित प्रश्न

Write the dual of the following statements: (p ∨ q) ∧ T


Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )


If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧  ∼ r ) ∧ ( ∼ q ∧ s )


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. 


Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]


Express the following statement in symbolic form and write its truth value.
"If 4 is an odd number, then 6 is divisible by 3."


Use the quantifiers to convert the following open sentence defined on N into true statement
5x - 3 < 10


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


Write the negation of the Following Statement :
∀ y ∈  N, y2 + 3 ≤ 7


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 ∨ q) ∧ ∼ (p ∧ q)]


Using the truth table prove the following logical equivalence.

p → (q ∧ r) ≡ (p ∧ q) (p → r)


Using the truth table proves the following logical equivalence.

∼ (p ↔ q) ≡ (p ∧ ∼ q) ∨ (q ∧ ∼ p)


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

(p ∧ q) → (q ∨ p)


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

(p → q) ↔ (∼ p ∨ q)


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

(p ↔ q) ∧ (p → ∼ q)


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

(p ∧ ∼ q) ↔ (p → q)


Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .


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)


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

(~ q ∧ p) ∧ (p ∧ ~ p)


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

(p ∧ ~ q) → (~ p ∧ ~ q)


Prove that the following statement pattern is a tautology.

(p → q) ↔ (~ q → ~ p)


Prove that the following statement pattern is a tautology.

(~p ∧ ~q ) → (p → q)


Prove that the following statement pattern is a tautology.

(~ p ∨ ~ q) ↔ ~ (p ∧ q)


Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ ~p


Prove that the following statement pattern is a contradiction.

(p → q) ∧ (p ∧ ~ q)


Fill in the blanks :

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


Show that the following statement pattern is contingency.

(p → q) ↔ (~ p ∨ q)


Show that the following statement pattern is contingency.

p ∧ [(p → ~ q) → q]


Using the truth table, verify

p → (p → q) ≡ ~ q → (p → q)


Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p


Write the dual of the following:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r


Write the dual statement of the following compound statement.

13 is prime number and India is a democratic country.


Write the dual statement of the following compound statement.

Radha and Sushmita cannot read Urdu.


Write the negation of the following statement.

Some continuous functions are differentiable.


Using the rules of negation, write 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 ∨ 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) ∧ ~(q ∧ ~p)


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

If a man is bachelor, then he is happy.


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

p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]


Write the dual of the following.

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)


Write the dual of the following.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)


Write the dual of the following.

~(p ∨ q) ≡ ~p ∧ ~q


Choose the correct alternative:

If p is any statement, then (p ˅ ~p) is a


Choose the correct alternative:

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


The contrapositive of p → ~ q is ______


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`


If p → (∼p v q) is false, then the truth values of p and q are respectively


Which of the following is not true for any two statements p and q?


Using truth table verify that:

(p ∧ q)∨ ∼ q ≡ p∨ ∼ q


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

[(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)


The statement pattern (∼ p ∧ q) is logically equivalent to ______.


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

(∼p ∧ ∼q) → (p → q)


The converse of contrapositive of ∼p → q is ______.


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

(p ∧ q) → (q ∨ p)


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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