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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

Write the converse and contrapositive of the following statements. “If a function is differentiable then it is continuous” - Mathematics and Statistics

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प्रश्न

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

बेरीज

उत्तर

Let p: A function is differentiable,

q: It is continuous.

∴ The symbolic form of the given statement is p → q.

Converse: q → p

i.e. If a function is continuous then it is differentiable

Contrapositive: ~q → ~p

i.e. If a function is not continuous then it is not differentiable.

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पाठ 1.1: Mathematical Logic - Short Answers I

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

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)


Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`


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)]


Show that the following statement pattern in contingency : 

(~p v q) → [p ∧ (q v ~ q)] 


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.


State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.


By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency.  (p →  q) ∧  (p ∧ ~ q ).


Using the truth table prove the following logical equivalence.

p ↔ q ≡ ∼ [(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 ∨ r)] ↔ ∼ [p → (q → r)]


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


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

(p → q) ∨ (q → p)


Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)


Prepare truth table for (p ˄ q) ˅ ~ r

(p ∧ q) ∨ ~ r


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

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) ↔ ~ (p ∧ q)


Prove that the following statement pattern is a contradiction.

(p ∨ q) ∧ (~p ∧ ~q)


Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ ~p


If p is any statement then (p ∨ ∼p) is a ______.


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 → q) ↔ (~ p ∨ q)


Show that the following statement pattern is contingency.

(p → q) ∧ (p → r)


Using the truth table, verify.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)


Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p


Prove that the following pair of statement pattern is equivalent.

p → q and ~ q → ~ p and ~ p ∨ q


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) ≡ ~ p ∨ ~ q


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.

Karina is very good or everybody likes her.


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 → r) ∧ q


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

~(p ∨ q) → r


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)


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)


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)]


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 ∧ q ≡ [(p ∨ q)] ∧ ~p


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

If a man is bachelor, then he is happy.


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

If I do not work hard, then I do not prosper.


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

p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]


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 dual of the following.

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)


Write the dual of the following.

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)


Write the dual of the following

(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (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 ______


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`


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


Which of the following is not equivalent to p → q.


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


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


Using truth table verify that:

(p ∧ q)∨ ∼ q ≡ p∨ ∼ q


Write the negation of the following statement:

(p `rightarrow` q) ∨ (p `rightarrow` r)


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) → (q ∨ p)


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