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Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p. 

So, The negation of the statement is “The sun is not cold.”

We know that a conditional statement is logically equivalent to its contrapositive.

Contrapositive: The weather will not be cold, if it does not snow.

A statement is a declarative sentence if it is either true or false but not both.

Here, the given sentence “Where is your bag” is a question.

Hence, it is not a statement.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “Number 7 is prime and odd” are,

p:Number 7 is prime.

q: Number 7 is odd.

A statement is a declarative sentence if it is either true or false but not both.

The given sentence “sky is Red” is false.

Hence, it is a false statement

(i) p: Number 7 is prime.

q: Number 7 is odd.

(ii) P: Chennai is in India.

q: Chennai is capital of Tamil Nadu.

(iii) p. 100 is divisible by 3.

q: 100 isdivisibleby 11.

r: 100 ¡s divisible by 5.

(iv) p. Chandigarh is capital of Haryana.

q: Chandigarb is capital of UP

(v) p: √7 is a rational number.

q: √7 is an irrational number.

(vi) p: 0 is less than every positive integer.

q: O is less than every negative integer.

(vii) p: Plants use sunlight for photosynthesis.

q: Plants use water for photosynthesis.

q:- Plants use carbon dioxide for photosynthesis.

(viii) p: Two lines in a plane intersect at one point.

q: Two lines ¡n a plane are parallel.

(ix) p: A rectangle is a quadrilateral.

q. A rectangle is a 5-sided polygon.

(i) Here component statements are:

p: 57 is divisible by 2. [false]

q: 57 is divisible by 3. [true]

Given compound statement is of the form ‘pvq’.

Since, the statement ‘pvq’ has the truth value T whenever either p or q or both have the truth value T.

So, it is true statement as 57 is divisible by 3.

(ii) Here component statements are: p: 24 is multiple of 4. q: 24 is multiple of 6.

Given compound statement is of the form ‘p ^ q’

Since, the statement ‘p A q’ has the truth value T whenever bothp and q have the truth value T.

So, it is a true statement as 24 is divisible by 4 and 6.

(iii) Here component statements are:

p: All living things have two eyes. [false]

q: All living things have two legs. [false]

Given compound statement is of the form ‘p ^q’

It is a false statement. Since ‘p ^ q’ has truth value F whenever either p or q or both have the truth value F

(iv) Here component statements are:

p: 2 is an even number. [true]

q: 2 is a prime number. [true]

Given compound statement is of the form ‘p ^ q’.

It is a true statement. Since ‘p ^ q’ has truth value T whenever both p and q or both have the truth value T.

(i) Here, the connecting word is ‘and’.

The component statements are as follows.

p: All rational numbers are real.

q: All real numbers are not complex.

(ii) Here, the connecting word is ‘or’.

The component statements are as follows.

p: Square of an integer is positive.

q: Square of an integer is negative.

(iii) Here, the connecting word is ‘and’. 

The component statements are as follows.

p: The sand heats up quickly in the sun.

q: The sand does not cool down fast at night.

(iv) Here, the connecting word is ‘and’.

The component statements are as follows.

p: x = 2 is a root of the equation 3x² – x – 10 = 0

q: x = 3 is a root of the equation 3x² – x – 10 = 0

(i) The components of the compound statement are: 

P: The sky is blue. 

Q: The grass is green. 

(ii) The components of the compound statement are: 

P: The earth is round. 

Q: The sun is cold. 

(iii) The components of the compound statement are: 

P: All rational number is real. 

Q: All real number are complex. 

(iv) The components of the compound statement are: 

P: 25 is multiple of 5. 

Q: 25 is multiple of 8.

(i) Here, “or” is exclusive because it is not possible for the Sun to rise and the moon to set together.

(ii) Here, “or” is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) Here, “or” is exclusive because all integers cannot be both positive and negative.

(i) The number 17 is not prime.

(ii) 2 + 7≠6.

(iii) Violets are not blue.

(iv) √5 is not a rational number.

(v) 2 is a prime number.

(vi) Every real number is not an irrational number.

(vii) Cow does not have four legs.

(viii) A leap year does not have 366 days.

(ix) There exist similar triangles which are not congruent.

(x) Area of a circle is not same as the perimeter of the circle

(i)The negation of the first statement is “the number x is a rational number”.

This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.

Therefore, the given statements are negations of each other.

(ii) The negation of the first statement is “the number x is not a rational number”. This means that the number x is an irrational number, which is the same as the second statement. Therefore, the given statements are negations of each other.

(i) In the given statement

“Students can take Hindi or Sanskrit as their third language” an exclusive “OR” is used because a student cannot take both Hindi and Sanskrit as the third language. 

(ii) In the given statement “To entry a country, you need a passport or a voter registration card” an inclusive “OR” is used because 

A person can have both a passport and a voter registration card to enter a country. 

(iii) In the given statement “A lady gives birth to a baby boy or a baby girl.” An exclusive “OR” is used because A lady cannot give birth to a baby who is both a boy and a girl. 

(iv) In the given statement “To apply for a driving license, you should have a ration card or a passport” an inclusive “OR” is used because 

A person can have both a ration card and passport to apply for a driving license.

(i) The components of the compound statement are:

P: To get into a public library children need an identity card.

Q: To get into a public library children need a letter from the school authorities.

Both P and Q are true.

Hence, the compound statement is true.

(ii) The components of the compound statement are:

P: All rational number is real.

Q: All real numbers are not complex.

P is true and Q is false then P and Q both are False.

Hence, the compound statement is False

(iii) The components of the compound statement are:

P: Square of an integer is positive.

Q: Square of an integer is negative.

Both P and Q are true.

Hence, the compound statement is True.

(iv) The components of the compound statement are:

P: x=2 is a root of the equation 3x² – x – 10 = 0

Q: x = 3 is a root of the equation 3x² – x – 10 = 0

P is true, but Q is false then P and Q both are False.

Hence, the compound statement is False.

(v) The components of the compound statement are:

P: The sand heats up quickly in the sun.

Q: The sand does not cool down fast at night.

P is false and Q is also false then P and Q both are False.

Hence, the compound statement is False.

(i) The component statements are as follows.

p: Number 3 is prime. 

q: Number 3 is odd.

Both the statements are true.

(ii) The component statements are as follows.

p: All integers are positive. 

q: All integers are negative.

Both the statements are false.

(iii) The component statements are as follows.

p: 100 is divisible by 3. 

q: 100 is divisible by 11. 

r: 100 is divisible by 5.

Here, the statements, p and q, are false and statement r is true.

Negation of statement p is “not p.” The negation of p is symbolized by “~p.” The truth value of ~p is the opposite of the truth value of p.

The negation of the statement is “The number 17 is not prime”.

Here the statement is “If p, then q”

i.e p→ q

Contrapositive of the statement p→ q is (~q)→ (~p)

Therefore,

If ~q, then ~p

Hence, the correct option is (C)

Given q : 131 is a multiple of 3 or 11

Let, P: 131 is a multiple of 3.

Q: 131 is a multiple of 11.

Here, P is false and Q is False

Therefore, P ⋁ Q is False

Hence, q is not valid

Let p: x² is not even

q: x is not even

So, The converse of the statement p→ q is q→ p

Therefore,

If x is not even, then x² is not even.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “The number 100 is divisible by 3, 11 and 5”.

p: 100 is divisible by 3.

q: 100 is divisible by 11.

r: 100 is divisible by 5.

p: 125 is divisible by 5 and 7

Let, q: 125 is divisible by 5.

r: 125 is divisible 7.

Here, q is true and r is false.

Therefore, q ᴧ r is False

Hence, p is not valid.

(i) if I become a doctor, then I will open hospital.

p : I become a doctor.

q : I will open a hospital.

~(p ⇒ q) ≡ will open hospital

∴ The negation of the statement is-

‘I will become a doctor and I will not open a hospital.

(ii) If 2 + 3 = 5, then is an odd number.

p : 2 + 3 = 5

q : 5 is an odd number

~(p ⇒ q) ≡ p ∩ ~ q

∴ The negation of the statement is-

“2 + 3 = 5 and 5 is an even number.

The given statement is compound statement then components are,

P: 6 is divisible by 2

~p: 6 is not divisible by 2

q: 6 is divisible by 3

~q: 6 is not divisible by 3.

(p ᴧ q)= 6 is divisible by 2 and 3.

~ (p ᴧ q) = ~p v ~q= 6 is neither divisible by 2 nor 3

In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: You do not study

q: you will fail.

Therefore,

If you do not study, then you will fail.”

In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: The number is odd.

q: The square of odd number is odd.

Therefore,

If the number is odd, then its square is odd number.

In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: Take the dinner

q: you will get sweet dish

Therefore,

If take the dinner, then you will get sweet dish.

(i) You get an A grade if and only if you do all your homework regularly.

(ii) You watch television if and only if your mind is free.

(i) p ⟷ q: The unit digit of on integer is zero, if and only if it is divisible by 5.

(ii) p ⟷ q: A natural number is odd if and only if it is not divisible by 2.

(iii) p ⟷q: A triangle is an equilateral triangle if and only if all three sides of triangle are equal.

Correct option (b)  p→(p v q)

Explanation:

p→ (q→p) = ~ p v (q → p)

= ~ p v (~q v p) since p v ~ p is always true

= ~ p v p v q = p→ (p v q)

We know that p→ q is same as p only if q.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “Chandigarh is the capital of Haryana and U.P.” are,

p: Chandigarh is the capital of Haryana and U.P

q: Chandigarh is the capital of U.P

Let p:Chandigarh is the capital of Punjab

q: Chandigarh in India.

~p: Chandigarh is not Capital of Punjab

~q: Chandigarh is not in India.

Since,

If (~q), then (~p)

Therefore,

If chandigarh is not in India, then Chandigarh is not the capital is not the capital of Punjab.

(c) ‘p → q is same as ‘p only if q’.

(b) Let p: x² is not even.

and q: x is not even.

Converse of the statement p →q is q → p. i.e.,

If x is not even, then x² is not even.

The solution of above compound statements in terms of p and q are given below :

(i)  p ∨ q 

(ii)  p  ∧ ⌉q 

(iii)  ⌉(p ∧  q) 

(iv)  ⌉p ∨ ⌉q 

(v)  ⌉p ∧ q

(vi)  ( ⌉p ∨ ⌉q)

(vii)   p ∨ q 

(a) Let p: Chandigarh is capital of Punjab. and q: Chandigarh is in India.

~ p: Chandigarh is not capital of Punjab.

~q: Chandigarh is not in India.

Contra positive of the statement p → q

if (~q), then (~p).

It Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

(c) Letp: 7 is greater than 5.

and q: 8 is greater than 6.

∴P→ q

~p: 7 is not greater than 5.

~q: 8 is not greater than 6.

(~q) → (~p) i.e., if 8 is not greater than 6, then 7 is not greater than 5.

(i) The sun is a star is a statement.

It is a scientifically proven fact, therefore this sentence is always true.

(ii) An irrational number is any number which cannot be expressed as a fraction of two integers.

Here, √7 cannot be expressed as a fraction of two integers, so √7 is an irrational number.

Therefore, “√7 is an irrational number” is a statement, and it is true.

(iii) Sum of 5 and 6 = 5 + 6 = 11 > 10

Sum of 5 and 6 is 11, which is greater than 10.

Therefore, “The sum of 5 and 6 is less than 10” is a statement, but not true.

(iv) The sentence ‘Go to your class’ is an order.

This is an Imperative sentence. Hence it is not a statement.

(v) Ice is always cold is a statement.

It is scientifically proven the fact, therefore the sentence is always true.

(vi) The sentence ‘Have you ever seen the Red Fort?

This is an interrogative sentence. Hence not a statement.

(vii) ‘Every relation is a function’ is a statement.

There are relations which are not functions.

Therefore, the sentence is false.

(viii) ‘The sum of any two sides of a triangle is always greater than the third side’

It is a statement and mathematically proven result.

Hence the statement is true.

(ix) ‘May God bless you!’ is an exclamation sentence. Hence it is not a statement.

(i) If x ≠ 3, then x ≠y or y ≠ 3.

(ii) If n is not an integer, then it is not a natural number.

(iii) If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) If x is not positive integer, then either x or y is not negative integer.

(v) If natural number n is not divisible by 2 or 3, then n is not divisible by 6.

(vi) The weather will not be cold, if it does not snow.

(vii) lf x² is not less than 1, thenx is not a real number such that 0 <x <1.

(i) The sun is a star is a statement. It is a scientifically proven fact that the sun is a star and, therefore this sentence is always true. Hence it is a statement, and it is true.

 Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(ii) √7 is an irrational number. An irrational number is any number which cannot be expressed as a fraction of two integers. √7 cannot be expressed as a fraction of two integers, so √7 is an irrational number; therefore the sentence is always true. Hence it is a statement, and it is true. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(iii) The sentence is true because the sum of 5 and 6 is not less than 10. Sum of 5 and 6 is 11, which is not less than 10. Hence it is a statement. The statement is true. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(iv) This sentence ‘Go to your class’ is an order. Hence it is not a statement. 

Note: A sentence which is in the form of an order, exclamation and question is not a statement. 

(v) Ice is always cold is a statement. It is scientifically proven the fact that ice is always cold and, therefore the sentence is always true. 

Hence it is a statement, and it is true. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(vi) The sentence ‘Have you ever seen the Red Fort? Is a question, hence it is not a statement. 

Note: A sentence which is in the form of an order, exclamation and question is not a statement. 

(vii) The sentence ‘Every relation is a function’ is a statement. There are relations which are not functions. Therefore the sentence is false. Hence it is a statement, and it is false. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(viii) The sentence ‘The sum of any two sides of a triangle is always greater than the third side’ is a statement. Because the sum of any two sides of the triangle is always greater than the third side. Hence the statement is true. 

(ix) The sentence ‘May God bless you!’ is an exclamation. Hence it is not a statement. 

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(i) The sentence ‘Paris is in France’ is a statement. Paris is located in France, so the sentence given is true, so it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(ii) The sentence ‘Each prime number has exactly two factors’ is a statement. It is a mathematically proven fact that each prime number has exactly two factors, so the given sentence is true. Hence it is a statement. The statement is true. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(iii) The sentence ‘The equation x² + 5|x| + 6 = 0 has no real roots.’ Is a statement. x² + 5|x| + 6 = 0 do not have real roots. 

Case 1: (x ≥ 0) 

|x| = x: (x ≥ 0) 

x² + 5|x| + 6 = 0 

x² + 5x + 6 = 0 

(x + 2) (x + 3) = 0 

x = -2 and x = -3 

But we assumed x ≥ 0. So it is a contradiction. 

Case 2: (x <0) 

|x| = x: (x <0) 

x² + 5|x| + 6 = 0 

x² - 5x + 6 = 0 

(x - 2) (x - 3) = 0 

x = 2 and x = 3 

But we assumed x < 0. So it is a contradiction. 

So, there are no real roots for the equation x² + 5|x| + 6 = 0 

So, the given sentence is true, and it is a statement. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(iv) The sentence ‘(2 + √3) is a complex number’ is a statement. 

A number which can be expressed in the form ‘a+ib’ is a complex number, (2 + √3) cannot be expressed in ‘a+ib’ form, so 2 + √3 is not a complex number. So the given sentence is a statement, and it is false. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(v) The sentence ‘Is 6 a positive integer?’ is a question, so it is not a statement. 

Note: A sentence which is in the form of an order, exclamation and question is not a statement. 

(vi) The sentence ‘The product of -3 and -2 is -6’ is a statement. 

Because, the product of -3 and -2 is 6 not -6, the given sentence is false. Hence the given sentence is a statement. This statement is false. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(vii) The sentence given is a statement. It is mathematically proven that the angles opposite to the equal sides of an isosceles triangle are equal. So the given sentence is true, and it is a statement. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(viii) The sentence ‘Oh! It is too hot’ is not a statement. It is an exclamation, and hot is subjective, it is not a fact, and it is an opinion. So, the given sentence is not a statement. 

Note: A sentence which is in the form of an order, exclamation and question is not a statement. 

(ix) The sentence ‘Monica is a beautiful girl’ is not a statement. The given sentence is an opinion; this can be true for some cases, false for some other case. So, the given sentence is not a statement. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both. 

(x) The given sentence is a statement. 

Because not every quadratic equation will have a real root. So the given sentence is false. It is a statement. This statement is false. 

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(i) The components of the compound statement are:

P: Delhi is in India.

Q: 2 + 2 = 4

Both P and Q are true.

Hence, the compound statement is True.

(ii) The components of the compound statement are:

P: Delhi is in England.

Q: 2 + 2 = 4

P is false, and q is true. So, both P and Q are false.

Hence, the compound statement is False.

(iii) The components of the compound statement are:

P: Delhi is in India.

Q: 2 + 2 = 5

P is true, and q is false. So, both P and Q are false.

Hence, the compound statement is False.

(iv) The components of the compound statement are:

P: Delhi is in England.

Q: 2 + 2 = 5

Both P and Q are false.

Hence, the compound statement is False.

(i) The sum of 3 and 7 is 10 or 11.

Let p: The sum of 3 and 7 is 10.

And q: The sum of 3 and 7 is 11.

First sentence is TRUE. Second sentence is FALSE.

Or used is ‘Exclusive or’.

(ii) (1 + i) is a real or a complex number.

Let q: (1 + i) is a real number.

And q: (1 + i) is a complex number.

First sentence is TRUE. Second sentence is FALSE.

Or used is ‘Exclusive or’.

(iii) Every quadratic equation has one or two real roots.

Let q: Every quadratic equation has one real root.

And q: Every quadratic equation has two real roots.

P and q both are False.

Given sentence is FALSE.

(iv) You are wet when it rains, or you are in a river.

Let p: You are wet when it rains.

And q: You are wet when you are in a river.

(p or q) is true.

Or used is ‘Inclusive or’ because you can get wet either it rains or when you are in the river.

(v) 24 is a multiple of 5 or 8.

Let p: 24 is a multiple of 5.

And q: 24 is a multiple of 8.

First sentence is FALSE. Second sentence is TRUE.

Or used is ‘Exclusive or’.

(vi) Every integer is rational or irrational.

Let p: Every integer is rational.

And q: Every integer is irrational.

(p or q) is true.

‘Or’ used is ‘Exclusive or’.

(vii) For getting a driving license you should have a ration card or a passport.

Let p: For getting a driving license you should have a ration card.

And q: For getting a driving license you should have a passport.

(p or q) is true.

Or used is ‘Inclusive or’, because you can get a driving license with ration card or with passport or when they have both.

(viii) 100 is a multiple of 6 or 8.

Let p: 100 is a multiple of 6.

And q: 100 is a multiple of 8.

(p or q) is FALSE.

(ix) Square of an integer is positive or negative.

Let p: Square of an integer is positive.

And q: Square of an integer is negative.

(p or q) is FALSE.

(x) Sun rises or Moon sets.

Let p: Sun rises.

And q: Moon sets.

(p or q) is TRUE.

Here, Or used is ‘Exclusive or’.

(i) Given: The open sentence x + 2 < 10 is defined on N.

Here N: {1, 2, 3, 4, 5, 6, 7, 8, …….…}

At x = 1 ⇒ x + 2 = 3 < 10

At x = 2 ⇒ x + 2 = 4 < 10

At x = 3 ⇒ x + 2 = 5 < 10

At x = 4 ⇒ x + 2 = 6 < 10

At x = 5 ⇒ x + 2 = 7 < 10

At x = 6 ⇒ x + 2 = 8 < 10

At x = 7 ⇒ x + 2 = 9 < 10

At x = 8 ⇒ x + 2 = 10

x = {1, 2, 3, 4, 5, 6, 7} satisfies x + 2 <10.

So, the truth set of open sentence x + 2 < 10 defined on N : {1, 2, 3, 4, 5, 6, 7}

(ii) The open sentence x + 5 < 4 is defined on N.

Here N: {1, 2, 3, 4, 5, 6, 7, 8, …….…}

At x = 1 ⇒ 1 + 5 = 6 > 4

So, the truth set of open sentence x + 5 < 4 defined on N is an empty set, {}.

(iii) The open sentence x + 3 > 2 is defined on N.

Here N: {1, 2, 3, 4, 5, 6, 7, 8, …….…}

At x = 1 ⇒ x + 3 = 4 > 2

At x = 2 ⇒ x + 3 = 5 > 2

At x = 3 ⇒ x + 3 = 6 > 2

At x = 4 ⇒ x + 3 = 7 > 2

At x = 5 ⇒ x + 3 = 8 > 2

At x = 6 ⇒ x + 3 = 9 > 2

At x = 7 ⇒ x + 3 = 10 > 2

And so on…

x = {1, 2, 3, 4, 5, 6, 7….} satisfies x + 3 > 2.

So, the truth set of open sentence x + 3 > 2 defined on N is an infinite set i.e. {1, 2, 3, 4, 5, 6, 7, ………}

(i) If the rectangle ‘S’ is rhombus, then it is square.

(ii) It tomorrow is Tuesday, then today is Monday.

(iii) If you must visit Taj Mahal, you go to Agra.

(iv) If the triangle is right angle, then the sum of squares of two sides of a triangle is equal to the square of third side.

(v) If the triangle is equilateral, then all three angles of triangle are equal.

(vi) If 2x = 3y, thenx : y = 3 : 2

(vii) If the opposite angles of a quadrilateral are supplementary, then S is cyclic.

(viii) If x is neither positive nor negative, then x is 0.

(ix) If the ratio of corresponding sides of two triangles are equal, then triangles are similar

(i) 57 is divisible by 2 or 3.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “57 is divisible by 2 or 3” are

p: 57 is divisible by 2.

q: 57 is divisible by 3.

Now, the given compound statement is in the form of P V Q, that has truth value T whenever either P or Q or both will true.

Hence, the given statement is true.

(ii) 24 is a multiple of 4 and 6.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “24 is a multiple of 4 and 6” are

p: 24 is a multiple of 4.

q: 24 is a multiple of 6.

Now, both the component p and q are true. As 24 is a multiple of both 4 and 6

Hence, the given statement is true.

(iii) All living things have two eyes and two legs.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “All living things have two eyes and two legs” are

p: All living things have two eyes.

q: All living things have two legs

Now, the given compound statement is in the form of P ⋂ Q that has truth value True

Only when, both the components will be true.

Here,

“All living things have two eyes” is False

“All living things have two legs” is False

Hence, the given statement is False.

(iv) 2 is an even number and a prime number.

A compound statement is a combination of two statements (Components).

So, the components of the given statement “2 is an even number and a prime number” are

p: 2 is an even number.

q: 2 is an prime number.

Now, the given compound statement is in the form of P ⋂ Q that has truth value True

Only when, both the components will be true. Here,

“2 is an even number” is true

“2 is an prime number” is true

Hence, the given statement is true.

(i) The components of the compound statement are: 

P: Delhi is in India. 

Q: 2+2 = 4 

Here, P and Q are true then P or Q is True. 

Hence, The compound statement is True. 

(ii) The components of the compound statement are: 

P: Delhi is in England. 

Q: 2+2 = 4 

Here, P is false, and q is true . So, P and Q must be false. 

Hence, The compound statement is False. 

(iii) The components of the compound statement are: 

P: Delhi is in India. 

Q: 2+2 = 5 

Here, P is True, and q is False . So, P and Q must be false. 

Hence, The compound statement is False. 

(iv) The components of the compound statement are: 

P: Delhi is in England. 

Q: 2+2 = 5 

Here, P and q are False . So, P and Q must be false. 

Hence, The compound statement is False.

Given: A = {2, 3, 5, 7}

(i) ∃ x ∈ A such that x + 3 > 9.

We have to check whether there exists ‘x’ which belongs to ‘A’, such that x + 3 > 9.

When x = 7 ∈ A,

x + 3 = 7 + 3 = 10 > 9

So, ∃ x ∈ A and x + 3 > 9.

So, the given statement is TRUE.

(ii) ∃ x ∈ A such that x is even.

We have to check whether there exists ‘x’ which belongs to ‘A’, such that x is even.

x = 2, is an even number and 2 ∈ A.

So, the given statement is TRUE.

(iii) ∃ x ∈ A such that x + 2 = 6.

We have to check whether there exists ‘x’ which belongs to ‘A’, such that x + 2 = 6.

At x = 2 ⇒ x + 2 = 4 ≠ 6

At x = 3 ⇒ x + 2 = 5 ≠ 6

At x = 5 ⇒ x + 2 = 7 ≠ 6

At x = 7 ⇒ x + 2 = 9 ≠ 6

None of the values satisfy the equation.

So, the given statement is FALSE.

(iv) ∀ x ∈ A, x is prime.

We have to check whether for all ‘x’ which belongs to ‘A’, such that x is a prime number.

All ‘x’ which belongs to A = {2, 3, 5, 7} is a prime number.

All are prime numbers.

So, the given statement is TRUE.

(v) ∀ x ∈ A, x + 2 < 10.

We have to check whether for all ‘x’ which belongs to ‘A’, such that x + 2 < 10.

A = {2, 3, 5, 7}

At x = 2 ⇒ x + 2 = 4 < 10

At x = 3 ⇒ x + 2 = 5 < 10

At x = 5 ⇒ x + 2 = 7 < 10

At x = 7 ⇒ x + 2 = 9 < 10

∀ x ∈ A, x + 2 < 10, is a TRUE statement.

(vi) ∀ x ∈ A, x + 4 ≥ 11.

We have to check whether for all ‘x’ which belongs to ‘A’, such that x + 4 ≥ 11.

A = {2, 3, 5, 7}

At x = 2 ⇒ x + 4 = 6 ≥ 11

At x = 3 ⇒ x + 4 = 7 ≥ 11

At x = 5 ⇒ x + 4 = 9 ≥ 11

At x = 7 ⇒ x + 4 = 11 ≥ 11

Only for x = 7, statement is true.

∀ x ∈ A, x + 4 ≥ 11, is a FALSE statement.

(i) If each of the angles of a rhombus measures 90°, then the rhombus is a square.

(ii) If a number is a multiple of 9, then the number is multiple of 3.

(iii) If you get a job, then your credentials are good.

(iv) If it rains, then the atmospheric humidity increases.

(v) If a number is a multiple of 6, then it is a multiple of 3.

(i) The components of the compound statement are: 

P: To get into a public library children need an identity card. 

Q: To get into a public library children need a letter from the school authorities. 

We know if P and Q are true then P and Q must also be true. 

Hence, The compound statement is true. 

(ii) The components of the compound statement are: 

P: All rational number is real. 

Q: All real numbers are not complex. 

We know, P is true and Q is False then P and Q is False. 

Hence, The compound statement is False 

(iii) The components of the compound statement are: 

P: Square of an integer is positive. 

Q: Square of an integer is negative. 

We know that, if P and Q are true then P or Q is True. 

Hence, The compound statement is True. 

(iv) The components of the compound statement are:

P: x=2 is a root of the equation 3x²-x-10=0 

Q: x=3 is a root of the equation 3x²-x-10=0 

Here, P is true, but Q is False then P and Q is False.

Hence, The compound statement is False. 

(v) The components of the compound statement are: 

P: The sand heats up quickly in the sun. 

Q: The sand does not cool down fast at night. 

Here, P is false and Q is also False then P and Q is False. 

Hence, The compound statement is False.