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Definition of Converse: 

A conditional statement is not logically equivalent to its converse. 

Definition of contrapositive: A conditional statement is logically equivalent to its contrapositive. 

Converse: If you feel thirsty, then it is hot outside. 

Contrapositive: If you do not feel thirsty, then it is not hot outside.

Definition of Converse: 

A conditional statement is not logically equivalent to its converse. 

Definition of contrapositive: 

A conditional statement is logically equivalent to its contrapositive. 

Converse: If I go to a beach, then it is a sunny day. 

Contrapositive: If I do not go to a beach, then it is not a sunny day.

(i) Statement p can be written as follows.

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is as follows.

If a positive integer has no divisors other than 1 and itself, then it is prime.

The contrapositive of the statement is as follows.

If positive integer has divisors other than 1 and itself, then it is not prime.

(ii)The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is as follows.

If I go to a beach, then it is a sunny day.

The contrapositive of the statement is as follows.

If I do not go to a beach, then it is not a sunny day.

(iii) The converse of statement r is as follows.

If you feel thirsty, then it is hot outside.

The contrapositive of statement r is as follows. If

you do not feel thirsty, then it is not hot outside.

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

By the properties of quadrilateral “Sum of opposite angles of a cyclic quadrilateral is180°.”

So, the given sentence is true.

Hence, it is a statement

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 “No policemen is thief”.

(i) The number x is not a rational number. 

= The number x is an irrational number. 

Since The statement “The number x is not a rational number.” is a negation of the first statement. 

(ii) The number x is not a rational number. 

= The number x is an irrational number. 

Since The statement “The number x is a rational number.” Is not a negation of the first statement.

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: 

p : For every positive real number x, the number (x – 1) is also positive. is 

~p : There exists a positive real number x, such that the number (x – 1) is not positive.

(i) p : For every positive real number x, the number (x – 1) is also positive.

The negation of the statement:

p: For every positive real number x, the number (x – 1) is also positive.

is

~p: There exists a positive real number x, such that the number (x – 1) is not positive.

(ii) q: For every real number x, either x > 1 or x < 1.

The negation of the statement:

q: For every real number x, either x > 1 or x < 1.

is

~q: There exists a real number such that neither x>1 or x<1.

(iii) r: There exists a number x such that 0 < x < 1.

The negation of the statement:

r: There exists a number x such that 0 < x < 1.

is

~r: For every real number x, either x ≤ 0 or x ≥ 1.

The negation of the statement is- “Zero is not a positive number”.

In the conjuction, we use “and” between two statement like p and q.

Hence, None of the given statements separated by and.

(b) Letp: 7 is greater than 8.

~p: 7 is not greater than 8

The negation of the given statement is false.

Since, It is false that a natural number is not greater than zero.

Hence, the correct option is (C)

B. 7 is not greater than 8.

If the statement is p then its negation is ~p, it means if p is true then ~p is false and vice versa.

Since, The negation of “7 is greater than 8 “ is “7 is not greater than 8”.

In the contrapositive, a conditional statement is logically equivalent to its contrapositive.

Since,

p: 7 is greater than 5

~p: 7 is not greater than 5

q: 8 is greater than 6

~q: 8 is not greater than 6

Therefore,

~p→ ~q = If 8 is not greater than 6, then 7 is not greater than 5.

(c) The false negation of the given statement is “It is false that a natural number is not

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For every natural number x, x + 1 is also a natural number.”

Quantifier is “For every”

Hence, ‘For every’ is quantifier.

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. The given sentence is a true declarative sentence. Hence, it is a true statement.

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all real numbers x with x > 3, x² is greater than 9.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

(i) Students can take Hindi or Sanskrit as their third language.

Exclusive ‘OR’, Since students can take either Sanskrit or Hindi but not both.

(ii) \(\sqrt{7}\) is a rational or an irrational number.

Here, ‘OR’ is exclusive, since \(\sqrt{7}\) is either rational or irrational but not both.

The given statement is compound statement then components are,

P: A triangle has 3 sides

~p: A triangle does not have 3 sides.

q: A triangle has 4 sides.

~q: A triangle does not have 4 side.

(p v q)= A triangle has either 3-sides or 4-sides.

~(p v q)=~p ᴧ ~q= A triangle has neither 3 sides nor 4 sides.

We know that a conditional statement is not logically equivalent to its converse.

Converse: If x is neither positive nor negative then x = 0

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 “All similar triangles are not congruent”.

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a triangle which is not an isosceles triangle.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all real numbers x and y, xy = yx.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

The quantifier here is “There exists”.

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a real number x such that x² + 1 = 0.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

We know that a conditional statement is not logically equivalent to its converse.

Converse: If the ratio of corresponding sides of two triangles are equal, then triangles are similar.

We know that a conditional statement is not logically equivalent to its converse.

Converse: If the triangle is right triangle, then the sum of the squares of two sides of a triangle is equal to the square of third side.

We know that a conditional statement is not logically equivalent to its converse.

Converse: if 2x = 3y then x: y = 3: 2

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a triangle which is not equilateral”

Quantifier is “There exist”

Hence, There exist is quantifier.

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

Contrapositive: If the triangle is not equilateral, then all three sides of the triangle are not equal.

We know that a conditional statement is not logically equivalent to its converse.

Converse: If the triangle is equilateral, then all three angles of the triangle are equal.

(i) A triangle is equilateral if and only if it is equiangular.

Let, p ∶ a triangle is equilateral.

q : A triangle is equiangular.

~ (p ⇒ q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p)

The negation of the statement is

“There exists either an equilateral triangle which is not equiangular or on equiangular triangle which is not equilateral.

(ii) Sets A and B are equal if and only if A ≤ B and B ≤ A.

Let, p : Sets A and B are equal.

q ∶ A ≤ B and B ≤ A.

~ (p(=)q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p)

The negation of the statement is-

Either A = B and (A ≤ B or B ≤ A.) or (A ≤ B and B ≤ A.) and ≠ B.”

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. The sentence “Listen to me, Ravi ! “ is an exclamatory sentence. Hence, it is not a statement.

a) “We won the match!”

The sentence “We won the match!” Is an exclamatory sentence.

∴ It is not a statement.

b) Can u get me cup of tea?

This sentence is an interrogative sentence.

∴ It is not a statement.

c) Please get me the book which is kept on the table.

This sentence is expressed either as a request or as a command, hence it is an imperative sentence.

∴ It is not a statement.

The three examples of sentences, which are not statements, are as follows.

(i) He is a doctor.

It is not evident from the sentence as to whom ‘he’ is referred to. Therefore, it is not a statement.

(ii) Geometry is difficult.

This is not a statement because for some people, geometry can be easy and for some others, it can be difficult.

(iii) Where is she going?

This is a question, which also contains ‘she’, and it is not evident as to who ‘she’ is. Hence, it is not a statement.

(i) It is a statement because it is always true. 

(ii) It is a statement because it is always true. 

(iii) It is a statement because it is false. 

(vi) It is not a statement because it is open sentence. 

(v) It is not a statement because it is an exclamation. 

(vi) It is not a statement 

∵ It is an order. 

(vii) It is not a statement 

∵ It is a question. 

(viii) Not a statement, 

∵ It is true on Thursday but not on other days. 

(ix) Not a statement 

∵ Sentence with variable pronoun like‘she’. 

(x) A statement ∵ It is false (statement) sentence. 

(xi) A statement ∵ It is a false sentence. 

(xii) Not a statement ∵ It may be true or false 

(xiii) A statement ∵ It is true 

(xiv) Not a statement ∵ This sentence is not always true 

(xv) A statement ∵ This sentence is always true, as is natural phenomenon. 

(xvi) Not a statement ∵ It is false sentence maximum number of days in a month can never exceed. 

(xvii) Not a statement ∵ It may be true or false for some people mathematics can be easy and some people mathematics can be difficult. 

(xviii) A statement∵ It is true sentence. 

Since 5 + 7 = 12 >10 (xix) 

Not a statement, 

∵ It is sometimes true and sometimes false. 

Since (2)²  = 4, even and (3)²  = 9, odd. (xx) 

Not a statement ∵ It may be true or false. Since square has equal length sides, rectangle has unequal length sides. 

(xxi) Not a statement, ∵ It is an order. 

(xxii) A statement ∵ It is false sentence. 

(xxiii) Not a statement ∵ which day is not mentioned 

(xxiv) A statement ∵ It is true sentence. 

(xxv) A statement ∵ It is always true.

(b) In ‘2 + 7 > 9 or 2 + 7 < 9’, or is the connective.

(d) Connective word is ‘and’.

(c) We have, p: Rajesh or Rajni lived in Bangalore.

and q: Rajesh lived in Bangalore.

r: Rajni lived in Bangalore.

~q: Rajesh did not live in Bangalore.

~r. Rajni did not live in Bangalore.

~ (q v r): Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

(b) Now, p: Plants take in CO₂ and give out O₂ .

Let q: Plants take in CO₂ .

r: Plants give out O₂ .

~q: Plants do not take in CO₂ .

~r: Plants do not give out O₂ .

~(q ∧ r): Plants do not take in CO₂ or do not give out O₂ .

(i) The compound statement ‘if p then q’ is implication of p and It is denoted by p → q or p ⇒ q. (read : p implication q)

Rule:

Note: If ‘p’ and then ‘q’ is small following:

• p ⇒ q (i.e., p implies q) 
• p is sufficient condition for q
• p only if q
• q is necessary condition for p
• ~q implies ~p (i.e., ~q ⇒ ~p) 
• The compound statement ‘p if and only if q’ is double implication of p and It is denoted by p ⇔ q (read: p double implication q) 
• Rule: 

Note: ‘p if and only if q’ is same as the following.

• p ⇔ q
• p if and only if q 
• q if and only if p
• p is necessary and sufficient condition for q and vice-versa

(i) Chennai is not the capital of Tamil Nadu.

(ii) √2 is a complex number.

(iii) All triangles are equilateral triangles.

(iv) The number 2 is not greater than 7.

(v) Every natural number is not an integer.

(i) This sentence is incorrect because the maximum number of days in a month is 31. Hence, it is a statement.

(ii) This sentence is subjective in the sense that for some people, mathematics can be easy and for some others, it can be difficult. Hence, it is not a statement.

(iii) The sum of 5 and 7 is 12, which is greater than 10. Therefore, this sentence is always correct. Hence, it is a statement.

(iv) This sentence is sometimes correct and sometimes incorrect. For example, the square of 2 is an even number. However, the square of 3 is an odd number. Hence, it is not a statement.

(v) This sentence is sometimes correct and sometimes incorrect. For example, squares and rhombus have sides of equal lengths. However, trapezium and rectangles have sides of unequal lengths. Hence, it is not a statement.

(vi) It is an order. Therefore, it is not a statement.

(vii) The product of (–1) and 8 is (–8). Therefore, the given sentence is incorrect. Hence, it is a statement.

(viii) This sentence is correct and hence, it is a statement.

(ix) The day that is being referred to is not evident from the sentence. Hence, it is not a statement.

(x) All real numbers can be written as a × 1 + 0 × i. Therefore, the given sentence is always correct. Hence, it is a statement.

The converse of the statement is –

“if a number is divisible by 5, then it is divisible by 10.”

The converse of the statement is-

“if x is an integer, then it is a natural number.”

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 “No Even integer is prime.”

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 “I will go to school.”

The given sentence is a compound statement in which components are

p: A number is divisible by 2

q: A number is divisible by 3

Now, it can be represent in symbolic function as,

p V q: A number is either divisible by 2 or 3.

The given sentence is a compound statement in which components are

p: x is an odd integer

q: x+1 is an odd integer

Now, it can be represent in symbolic function as,

p V q: Either x or x + 1 is an odd integer.