Balbharati solutions for माठेमटिक्स अँड स्टॅटिस्टिक्स १ (कॉमर्स) [इंग्रजी] इयत्ता १२ महाराष्ट्र राज्य मंडळ chapter 1 - Mathematical Logic [Latest edition]

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

A triangle has ‘n’ sides.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

The sum of interior angles of a triangle is 180°

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

You are amazing!

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Please grant me a loan.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

`sqrt(-4)` is an irrational number.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

x² − 6x + 8 = 0 implies x = −4 or x = −2.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

He is an actor.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Did you eat lunch yet?

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Have a cup of cappuccino.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

(x + y)² = x² + 2xy + y² for all x, y ∈ R. 

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Every real number is a complex number.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

1 is a prime number.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

With the sunset the day ends.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

1 ! = 0

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

3 + 5 > 11

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

The number π is an irrational number.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

x² - y² = (x + y)(x - y) for all x, y ∈ R.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

The number 2 is the only even prime number.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Two co-planar lines are either parallel or intersecting.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

The number of arrangements of 7 girls in a row for a photograph is 7!.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Give me a compass box.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Bring the motor car here.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

It may rain today.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

If a + b < 7, where a ≥ 0 and b ≥ 0 then a < 7 and b < 7.

State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

Can you speak in English?

Express the following statement in symbolic form.

e is a vowel or 2 + 3 = 5

Express the following statement in symbolic form.

Mango is a fruit but potato is a vegetable.

Express the following statement in symbolic form.

Milk is white or grass is green.

Express the following statement in symbolic form.

I like playing but not singing.

Express the following statement in symbolic form.

Even though it is cloudy, it is still raining.

Write the truth value of the following statement.

Earth is a planet and Moon is a star.

Write the truth value of the following statement.

16 is an even number and 8 is a perfect square.

Write the truth value of the following statement.

A quadratic equation has two distinct roots or 6 has three prime factors.

Write the truth value of the following statement.

The Himalayas are the highest mountains but they are part of India in the North East.

Write the negation of the following statement.

All men are animals.

Write the negation of the following statement.

− 3 is a natural number.

Write the negation of the following statement.

It is false that Nagpur is capital of Maharashtra

Write the negation of the following statement.

2 + 3 ≠ 5

Write the truth value of the negation of the following statement.

`sqrt5` is an irrational number.

Write the truth value of the negation of the following statement.

London is in England.

Write the truth value of the negation of the following statement.

For every x ∈ N, x + 3 < 8.

Write the following statement in symbolic form.

If triangle is equilateral then it is equiangular.

Write the following statement in symbolic form.

It is not true that “i” is a real number.

Write the following statement in symbolic form.

Even though it is not cloudy, it is still raining.

Write the following statement in symbolic form.

Milk is white if and only if the sky is not blue.

Write the following statement in symbolic form.

Stock prices are high if and only if stocks are rising.

Write the following statement in symbolic form.

If Kutub-Minar is in Delhi then Taj-Mahal is in Agra.

Find the truth value of the following statement.

It is not true that 3 − 7i is a real number.

Find the truth value of the following statement.

If a joint venture is a temporary partnership, then discount on purchase is credited to the supplier.

Find the truth value of the following statement.

Every accountant is free to apply his own accounting rules if and only if machinery is an asset.

Find the truth value of the following statement.

Neither 27 is a prime number nor divisible by 4.

Find the truth value of the following statement.

3 is a prime number and an odd number.

If p and q are true and r and s are false, find the truth value of the following compound statement.

p ∧ (q ∧ r)

If p and q are true and r and s are false, find the truth value of the following compound statement.

(p → q) ∨ (r ∧ s) 

If p and q are true and r and s are false, find the truth value of the following compound statement.

~ [(~ p ∨ s) ∧ (~ q ∧ r)]

If p and q are true and r and s are false, find the truth value of the following compound statement.

(p → q) ↔ ~(p ∨ q)

If p and q are true and r and s are false, find the truth value of the following compound statement.

[(p ∨ s) → r] ∨ ~ [~ (p → q) ∨ s]

If p and q are true and r and s are false, find the truth value of the following compound statement.

~ [p ∨ (r ∧ s)] ∧ ~ [(r ∧ ~ s) ∧ q]

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

Sunday is not holiday or Ram studies on holiday.

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

If Sunday is not holiday then Ram studies on holiday.

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

Sunday is a holiday and Ram studies on holiday.

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement:

p ↔ ~ q

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

~ (p ∨ q)

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

q → p

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

q ∧ ~ p

Use quantifiers to convert the following open sentences defined on N, into a true statement.

x² + 3x - 10 = 0 

Use quantifiers to convert the following open sentence defined on N, into a true statement.

3x - 4 < 9

Use quantifiers to convert the following open sentences defined on N, into a true statement.

n² ≥ 1

Use quantifiers to convert the following open sentences defined on N, into a true statement.

2n - 1 = 5

Use quantifiers to convert the following open sentences defined on N, into a true statement.

y + 4 > 6

Use quantifiers to convert the following open sentences defined on N, into a true statement.

3y - 2 ≤ 9

If B = {2, 3, 5, 6, 7} determine the truth value of ∀ x ∈ B such that x is prime number.

If B = {2, 3, 5, 6, 7} determine the truth value of
∃ n ∈ B, such that n + 6 > 12.

If B = {2, 3, 5, 6, 7} determine the truth value of
∃ n ∈ B, such that 2n + 2 < 4.

If B = {2, 3, 5, 6, 7} determine the truth value of
∀ y ∈ B, such that y² is negative.

If B = {2, 3, 5, 6, 7} determine the truth value of
∀ y ∈ B, such that (y - 5) ∈ N

Prepare truth tables for the following statement pattern.

p → (~ p ∨ q)

Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)

Prepare truth tables for the following statement pattern.

(p ∧ r) → (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)

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

~ p → (p → ~ q)

Prove that the following statement pattern is a tautology.

(p ∧ q) → 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 ∧ ~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)

Prove that the following statement pattern is a contradiction.

(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 ∨ q)

Show that the following statement pattern is contingency.

p ∧ [(p → ~ q) → 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 → (p → q) ≡ ~ q → (p → q)

Using the truth table, verify

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q.

Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Using the truth table, verify.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Prove that the following pair of statement pattern is equivalent.

p ↔ q and (p → q) ∧ (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 ∨ 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 dual statement of the following compound statement.

Karina is very good or everybody likes her.

Write the dual statement of the following compound statement.

Radha and Sushmita cannot read Urdu.

Write the dual statement of the following compound statement.

A number is a real number and the square of the number is non-negative.

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.

Write the negation of the following statement.

Some continuous functions are differentiable.

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

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

(~p ∧ q) ∧ (~q ∨ ~r)

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

"If it snows, then they do not drive the car"

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

Without using truth table, show that

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Without using truth table, show that

p ∧ [(~ p ∨ q) ∨ ~ q] ≡ p

Without using truth table, show that

~ [(p ∧ q) → ~ q] ≡ p ∧ q

Without using truth table, show that

~r → ~ (p ∧ q) ≡ [~ (q → r)] → ~ p

Without using truth table, show that

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Using the algebra of statement, prove that

[p ∧ (q ∨ r)] ∨ [~ r ∧ ~ q ∧ p] ≡ p

Using the algebra of statement, prove that

(p ∧ q) ∨ (p ∧ ~ q) ∨ (~ p ∧ ~ q) ≡ (p ∨ ~ q)

Using the algebra of statement, prove that (p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ (~ p ∧ q).

Represent the truth of the following statement by the Venn diagram.

Some hardworking students are obedient.

Represent the truth of the following statement by the Venn diagram.

No circles are polygons.

Represent the truth of the following statement by the Venn diagram.

All teachers are scholars and scholars are teachers.

Represent the truth of the following statement by the Venn diagram.

If a quadrilateral is a rhombus, then it is a parallelogram.

Draw a Venn diagram for the truth of the following statement.

Some share brokers are chartered accountants.

Draw a Venn diagram for the truth of the following statement.

No wicket keeper is bowler, in a cricket team.

Represent the following statement by the Venn diagram.

Some non-resident Indians are not rich.

Represent the following statement by the Venn diagram.

No circle is rectangle.

Represent the following statement by the Venn diagram.

If n is a prime number and n ≠ 2, then it is odd.

Which of the following is not a statement?

Choose the correct alternative :

Which of the following is an open statement?

Let p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Then, this law is known as ______.

Which of the following statements is false?

Choose the correct alternative :

For the following three statements
p : 2 is an even number.
q : 2 is a prime number.
r : Sum of two prime numbers is always even.
Then, the symbolic statement (p ∧ q) → ∼ r means.

Choose the correct alternative :

If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

The negation of the proposition “If 2 is prime, then 3 is odd”, is ______.

Choose the correct alternative :

The statement (∼ p ∧ q) ∨∼ q is

Choose the correct alternative:

Which of the following is always true?

Choose the correct alternative :

∼ (p ∨ q) ∨ (∼ p ∧ q) is logically equivalent to

If p and q are two statements then (p → q) ↔ (∼ q → ∼ p) is ______.

Choose the correct alternative :

If p is the sentence ‘This statement is false’ then

Choose the correct alternative :

Conditional p → q is equivalent to

Choose the correct alternative :

Negation of the statement “This is false or That is true” is

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

Fill in the blanks :

The statement q → p is called as the ––––––––– of the statement p → q.

Fill in the blanks :

Conjunction of two statement p and q is symbolically written as ______.

If p ∨ q is true then truth value of ∼ p ∨ ∼ q is ______.

Negation of “some men are animal” is ______.

Fill in the blanks :

Truth value of if x = 2, then x2 = − 4 is –––––––––.

Fill in the blanks :

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

Fill in the blanks :

p ↔ q is false when p and q have ––––––––– truth values.

Fill in the blanks :

Let p : the problem is easy. r : It is not challenging then verbal form of ∼ p → r is –––––––––.

Fill in the blanks :

Truth value of 2 + 3 = 5 if and only if − 3 > − 9 is –––––––––.

State whether the following statement is True or False :

Truth value of 2 + 3 < 6 is F.

State whether the following statement is True or False :

There are 24 months in year is a statement.

State whether the following statement is True or False :

p ∨ q has truth value F is both p and q has truth value F.

State whether the following statement is True or False:

The negation of 10 + 20 = 30 is, it is false that 10 + 20 ≠ 30.

State whether the following statement is True or False:

Dual of (p ∧ ∼ q) ∨ t is (p ∨ ∼ q) ∨ c.

State whether the following statement is True or False :

Dual of “John and Ayub went to the forest” is “John and Ayub went to the forest”.

State whether the following statement is True or False :

“His birthday is on 29th February” is not a statement.

State whether the following statement is True or False :

x² = 25 is true statement.

State whether the following statement is True or False :

Truth value of `sqrt(5)` is not an irrational number is T.

State whether the following statement is True or False :

p ∧ t = p.

Solve the following :

State which of the following sentences are statements in logic.
Ice cream Sundaes are my favourite.

Solve the following :

State which of the following sentences are statements in logic.
x + 3 = 8 ; x is variable.

Solve the following :

State which of the following sentences are statements in logic.
Read a lot to improve your writing skill.

Solve the following :

State which of the following sentences are statements in logic.
z is a positive number.

Solve the following :

State which of the following sentences are statements in logic.
(a + b)² = a² + 2ab + b² for all a, b ∈ R.

Solve the following :

State which of the following sentences are statements in logic.
(2 + 1)² = 9.

Solve the following :

State which of the following sentences are statements in logic.
Why are you sad?

Solve the following :

State which of the following sentences are statements in logic.
How beautiful the flower is!

Solve the following :

State which of the following sentences are statements in logic.
The square of any odd number is even.

Solve the following :

State which of the following sentences are statements in logic.
All integers are natural numbers.

Solve the following :

State which of the following sentences are statements in logic.
If x is real number then x2 ≥ 0.

Solve the following :

State which of the following sentences are statements in logic.
Do not come inside the room.

Solve the following :

State which of the following sentences are statements in logic.
What a horrible sight it was!

Which of the following sentence is a statement? In case of a statement, write down the truth value.

What is happy ending?

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The square of every real number is positive.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

Every parallelogram is a rhombus.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

a² − b² = (a + b) (a − b) for all a, b ∈ R.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

Please carry out my instruction.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The Himalayas is the highest mountain range.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

(x − 2) (x − 3) = x² − 5x + 6 for all x∈R.

Which of the following sentence is a statement? In case of a statement, write down the truth value.

What are the causes of rural unemployment?

Which of the following sentence is a statement? In case of a statement, write down the truth value.

0! = 1

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The quadratic equation ax² + bx + c = 0 (a ≠ 0) always has two real roots.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The Sun has set and Moon has risen.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

Mona likes Mathematics and Physics.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

3 is prime number if 3 is perfect square number.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

Kavita is brilliant and brave.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If Kiran drives the car, then Sameer will walk.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The necessary condition for existence of a tangent to the curve of the function is continuity.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

To be brave is necessary and sufficient condition to climb the Mount Everest.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

x³ + y³ = (x + y)³ if xy = 0.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The drug is effective though it has side effects.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If a real number is not rational, then it must be irrational.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

It is not true that Ram is tall and handsome.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

Even though it is not cloudy, it is still raining.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

It is not true that intelligent persons are neither polite nor helpful.

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If the question paper is not easy then we shall not pass.

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

Proof is lengthy and it is not interesting.

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

If proof is lengthy then it is interesting.

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

It is not true that the proof is lengthy but it is interesting.

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

It is interesting iff the proof is lengthy.

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

(p ∧ q) ∨ r

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

p → r

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

∼ p ∨ q

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.
p→(q ∨ r)

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

p → q

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

(p ∧ q) ∧ ∼ r

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

∼ (p ∨ q) ∧ r

Determine the truth value of the following statement.

4 + 5 = 7 or 9 − 2 = 5

Determine the truth value of the following statement.

If 9 > 1 then x² − 2x + 1 = 0 for x = 1

Determine the truth value of the following statement.

x + y = 0 is the equation of a straight line if and only if y² = 4x is the equation of the parabola.

Determine the truth value of the following statement.

It is not true that 2 + 3 = 6 or 12 + 3 =5

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

Stock prices are not high or stocks are rising.

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

Stock prices are high and stocks are rising if and only if stock prices are high.

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

If stock prices are high then stocks are not rising.

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

It is false that stocks are rising and stock prices are high.

Assuming the following statement.

p : Stock prices are high.

q : Stocks are rising.

to be true, find the truth value of the following.

Stock prices are high or stocks are not rising iff stocks are rising.

Rewrite the following statement without using conditional –
(Hint : p → q ≡ ∼ p ∨ q)

If price increases, then demand falls.

Rewrite the following statement without using conditional –
(Hint : p → q ≡ ∼ p ∨ q)

If demand falls, then price does not increase.

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

(p ∧ q) → ∼ p.

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

p ↔ (q → ∼ p)

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

(p ∧ ∼ q) ∨ (∼ p ∧ q)

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

∼ (p ∧ q) → ∼ (q ∧ p)

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

∼ [(p → q) ↔ (p ∧ ∼ q)]

Write the negation of the following.

If ∆ABC is not equilateral, then it is not equiangular.

Write the negation of the following.

Ramesh is intelligent and he is hard working.

Write the negation of the following.

An angle is a right angle if and only if it is of measure 90°.

Write the negation of the following.

Kanchanganga is in India and Everest is in Nepal.

Write the negation of the following.

If x ∈ A ∩ B, then x ∈ A and x ∈ B.

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

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

If 2 + 5 = 10, then 4 + 10 = 20.

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) ∨ (~ p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q)

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.

Rewrite the following statement without using the connective ‘If ... then’.

If a quadrilateral is rhombus then it is not a square.

Rewrite the following statement without using the connective ‘If ... then’.

If 10 − 3 = 7 then 10 × 3 ≠ 30.

Rewrite the following statement without using the connective ‘If ... then’.

If it rains then the principal declares a holiday.

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 ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)

Write the dual of the following.

~(p ∨ q) ≡ ~p ∧ ~q

Consider the following statements.

1. If D is dog, then D is very good.
2. If D is very good, then D is dog.
3. If D is not very good, then D is not a dog.
4. If D is not a dog, then D is not very good. 

Identify the pairs of statements having the same meaning. Justify.

Express the truth of the following statement by the Venn diagram.

All men are mortal.

Express the truth of the following statement by the Venn diagram.

Some persons are not politician.

Express the truth of the following statement by the Venn diagram.

Some members of the present Indian cricket are not committed.

Express the truth of the following statement by the Venn diagram.

No child is an adult.

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∃ x ∈ A, such that 3x + 2 > 9

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∀ x ∈ A, x² < 18.

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∃ x ∈ A, such that x + 3 < 11.

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∀ x ∈ A, x² + 2 ≥ 5.

Write the negation of the following statement.

7 is prime number and Tajmahal is in Agra.

Write the negation of the following statement.

10 > 5 and 3 < 8

Write the negation of the following statement.

I will have tea or coffee.

Write the negation of the following statement.

∀ n ∈ N, n + 3 > 9.

Write the negation of the following statement.

∃ x ∈ A, such that x + 5 < 11.