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The given statement can be written in five different ways as follows.

(i) A natural number is odd implies that its square is odd.

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd, it is necessary that its square is odd.

(iv) For the square of a natural number to be odd, it is sufficient that the number is odd.

(v) If the square of a natural number is not odd, then the natural number is not odd.

The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.

This is a false statement, since 25 is not a multiple of 8.

The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”. This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.

Let the statements,

p: Integer n is even

q: If n² is even

Let p be true. Then,

Let n = 2k

Squaring both the sides, we get,

n² = 4k²

n² = 2.2k²

n² is an even number.

So, q is true when p is true.

Hence, the given statement 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 “2 is a prime 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.

The negation of the statement is “Every real number is not an irrational 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.

The negation of the statement is √5 is not a rational 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.

The negation of the statement is “Cow does not have four legs”.

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 “Violets are not blue”.

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 “2 + 7 ≠ 6”.

(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.

Converse and contra positive are certain other statements which can be formed from a given statement with ‘if then’. 

• Converse: The converse of the statement if p, then q is defined as ‘if q then p’. i.e., the converse of a implication p ⇒ q is q ⇒ q 
• Contrapositive: The contrapositive of the statement if p, then q is defined as ‘if ~q, then ~p’ i.e., the contrapositive of a implication p ⇒ q is ~q ⇒ ~p 

Note: 

Inverse of a implication p ⇒ q is ~p ⇒ ~q

(i) ‘and’ is the connecting word. 

∴ component statements are 

p : all rational numbers are real 

q : all real numbers are not complex. 

(ii) Connecting word is ‘or’ component statements are 

p : square of an integer is positive 

q : square of an integer is negative 

(iii) Connecting word is ‘and’ component statements are 

p : sand heats up quickly in the sun 

q : sand does not cool down fast at night 

(iv) Connecting word is ‘and’Component statements are 

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

q : x = 3 is a root of 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.

As we know, a statement is a sentence which is either true or false but not both simultaneously.

(j) It is true statement.

(ii) It is true statement.

(iii) It is false statement.

(iv) It is false statement.

(y) It is false statement.

(vi) y +9 = 7

It is not considered as a statement, since the value of y is not given.

(vii) It is a question, so it is not a statement.

(viii) It is a true statement.

(ix) It is a true statement.

(x) It is a false statement.

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true. Hence, it is not a statement.

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. So, This sentence is always false, because there are non-empty sets whose intersection is empty. Hence, it is a statement.

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

The given sentence “Every set is an infinite set” is False.

Hence, it is a false statement

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

So, the given Expression “15 + 8 > 23” is false.

As the result of L.H.S will always equal to the result of R.H.S

Hence, it is a false statement

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. So, This sentence is always false, because there are sets which are not finite. Hence, it is a statement.

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

Here, y + 9 = 7 will be true for some value and it will be false for some value.

Like, at y= -2, the given expression is true and Y=1 or any other value, the expression is false.

Hence, it is not a statement

Compound statement is a statement which is made up of two or more statements using some connecting words like ‘and’ ‘or’, ‘if, then’, ‘if and only if etc. 

Note: Connecting words like ‘And’ ‘Or’, ‘If, then’, ‘If and only if are called connectives and each statement is called a component statement.

If p is a statement, then the negation of p is also a statement and is denoted ~p and read as ‘not p’. 

Note: While forming the negation of a statement, phrases like ‘It is not the case’ or ‘It is false that’ are also used.

(i) “Who is the Chancellor of your University”? 

Concept Used:

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

(ii) There are 31 days in a month 

Concept Used: 

A statement is an assertive (declarative) sentence if it is either true or false but not both. The given sentence is true for some particular months, and it is not true for others, so it may be true of false. Hence, It is a not statement. 

(iii) Hurray! We won the match. 

Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both. The given sentence is in an exclamatory sentence. 

Hence, It is a not statement.

(i) Listen to me, Ravi!

The sentence “Listen to me, Ravi! “Is an exclamatory sentence.

∴ It is not a statement.

(ii) Every set is a finite set.

This sentence is always false, because there are sets which are not finite.

∴ It is a statement.

(iii) Two non-empty sets have always a non-empty intersection.

This sentence is always false, because there are non-empty sets whose intersection is empty.

∴ It is a statement.

(iv) The cat ***** is black.

There are some cats which are black, and not black, So, the given sentence may or may not be true.

∴ It is not a statement.

(v) Are all circles round?

The given sentence is an interrogative sentence.

∴ It is not a statement.

(vi) All triangles have three sides.

The given sentence is a true declarative sentence.

∴ It is a true statement.

(vii) Every rhombus is a square.

This sentence is always false, because Rhombuses are not a square.

∴ It is a statement.

(viii) x² + 5|x| + 6 = 0 has no real roots.

Firstly, let us solve the given expression.

If x > 0,

x² + 5|x| + 6 = 0

x² + 5x + 6 = 0

x = -3 or x = -2

Since x > 0, the equation has no roots.

If x < 0,

x² + 5|x| + 6=0

x² – 5x + 6=0

Since x < 0, the equation has no real roots.

∴ It is a statement.

(ix) This sentences is a statement.

The statement is not indicating the correct value, hence we can say that value contradicts the sense of the sentence.

∴ It is not a statement.

(x) Is the earth round?

The given sentence is an interrogative sentence.

∴ It is not 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 “Not all birds sing.”

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 earth is not round.”

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

Converse: If tomorrow is Tuesday, then today is Monday.

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

Converse: If the rectangle R is rhombus, then it is square.

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

Contrapositive: If x²>1 then, x is not a real number such that 0<x<1.

In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: a, b and c are in AP

q: 2b=a + c

Therefore,

If a, b, c are in AP then 2b=a + c.

In the bi conditional statement, we use if and only if.

p: The unit digit of an integer is zero.

q: It is divisible by 5.

Then,

p ↔ q = Unit digit of an integer is zero if and only if it is divisible by 5.

In the conditional statement, expression is

If p, then q

Now,

The given statement p and q are

p: Any number is prime,

q: square of number is not prime.

Therefore,

If any number is prime, then its square is not prime”.

The converse statement is-

3 + 7 = 10, then 3 × 7 = 21.

In the conditional statement, expression is

If p, then q

Now,

In the given statement p and q are

p: An integer is divisible by 5

q: Unit digits of an integer are 0 or 5

Therefore,

If an integer is divisible by 5, then its unit digits are 0 or 5.

p: Rahul passed in Hindi.

q: Rahul passed in English. p ∧ q: Rahul passed in Hindi and English.

(ii) p: x is even integers. . q : y is even integers.

p ∧ q: x andy are even integers.

(iii) p: 2 is factor of 12. q: 3 is factor of 12. r: 6 is factor of 12.

p ∧ q ∧ r: 2, 3 and 6 are factors of 12

(iv) p: x is an odd integer.

q: (x + 1) is an odd integer. p v q: Either x or (x + 1) is an odd integer.

(v) p: A number is divisible by 2. q: A number is divisible by 3.

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

(vi) p: x = 2 is a root of 3x² – x – 10 = 0. q: x = 3 is a root of 3x² – x – 10 = 0

p v q: Either x = 2 or x = 3 is a root of 3x² – x – 10 = 0

(vii) p: Students can take Hindi as an optional paper. q: Students can take English as an optional paper.

p v q: Students can take Hindi or English as an optional paper.

(i) Let p: All rational numbers are real.

q: All rational numbers are complex.

~ p: All rational numbers are not real.

~ q ; All rational numbers are not complex.

Then, the negation of the given compound statement is:

~ (p ∧ q): All rational numbers are not real or not complex.

[~(p ∧ q) = ~p v ~q]

(ii) Let p: All real numbers are rationals. q: All real numbers are irrationals. Then, the negation of the given compound statement is:

~ (p v q): All real numbers are not rational and all real numbers are not irrational. [~(p v q) = ~p ∧ ~ q]

(iii) Let p ; x = 2 is root of quadratic equation x² – 5x + 6 = 0. q: x = 3 is root of quadratic equation x² – 5x + 6 = 0.

Then, the negation of the given compound statement is:

~ (p ∧ q) : x = 2 is not a root of quadratic equation x² – 5x + 6 = 0 or x = 3 is not a root of the quadratic equation x² – 5x + 6 = 0.

(iv) Let p: A triangle has 3-sides. q: A triangle has 4-sides.

Then, the negation of the given compound statement is:

~ (p v q): A triangle has neither 3-sides nor 4-sides.

(v) Let p: 35 is a prime number. q: 35 is a composite number.

Then, the negation of the given compound statement is:

~ (p v q): 35 is not a prime number and it is not a composite number.

(vi) Let p: All prime integers are even. q: All prime integers are odd.

Then, the negation of the given compound statement is given by

~(p v q): All prime integers are not even and all prime integers are not odd.

(vii) Let p:|x| is equal to x. q: |x| is equal to —x.

Then, the negation of the given compound statement is:

~ (p v q): |x| is not equal to JC and it is not equal to —x.

(viii) Let p: 6 is divisible by 2. 

q: 6 is divisible by 3.

Then, the negation of the given compound statement is:

~ (p∧q): 6 is not divisible by 2 or it is not divisible by 3

(i) If the number is odd number, then its square is odd number.

(ii) It you take the dinner, then you will get sweet dish.

(iii If you will not study, then you will fail.

(iv) If an integer is divisible by 5, then its unit digits are 0 or 5.

(v) If the number is prime, then its square is not prime.

(vi) If a, b and c are in AP, then 2b = a + c.

(i) New Delhi is not a city. 

(ii) It is false that everyone in Germany speaks German. 

(iii)  √7 is not rational 

(iv) Chennai is not the capital of Tamil Nadu 

(v)  √2 is a complex number 

(vi) The number 2 is not greater than 7 

(vii) Every natural number is not an integer.

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

• Note: In this chapter, statement means it is mathematically acceptable statement. 
• Note: If a sentence, which is vague or ambiguous then that sentence is not a statement. 
• Note: Open sentences are not statements. 
• Note: A sentence which contains where, why, what, he, she, other pronouns is not a statement. 

Mathematical statement is a sentence which, in a given context is either true or false but not both.

The basic unit involved in mathematical statement.

(i) The given statement is of the form “if q then r”.

q: All the angles of a triangle are equal. r: The triangle is an obtuse-angled triangle.

The given statement p has to be proved false.

For this purpose, it has to be proved that if q, then ∼r.

To show this, angles of a triangle are required such that none of them is an obtuse angle.

It is known that the sum of all angles of a triangle is 180°. Therefore, if all the three angles are equal, then each of them is of measure 60°, which is not an obtuse angle.

In an equilateral triangle, the measure of all angles is equal. However, the triangle is not an obtuse-angled triangle.

Thus, it can be concluded that the given statement p is false.

(ii) The given statement is as follows.q:

The equation x² – 1 = 0 does not have a root lying between 0 and 2.

This statement has to be proved false. To show this, a counter example is required.

Consider x² – 1 = 0 

x² = 1 

x = ± 1

One root of the equation x² – 1 = 0, i.e. the root x = 1, lies between 0 and 2.

Thus, the given statement is false.

Let  p : √7  is irrational 

Let us assume p is not true i.e..,  √7 is rational . 

⇒ √7 = a/b, where a and b are integers having no common factor.

⇒ 7 = a²/b²

⇒ a²  =7b² 

⇒ 7 divides a² 

⇒ 7 divides a 

⇒ a = 7c, for some integer c. 

⇒ a²  = 49c² 

⇒ 7b²  = 49c² 

⇒ b²  = 7c² 

⇒ 7 divides b² 

⇒ 7 divides b 

Thus, 7 is common factor of both a and b. This contradicts that a and b have no common factor. 

So, our assumption is wrong. 

Hence, √7  is irrational is true.

Contradiction statement: √3 is a rational number.

Proof:

If √3 is a rational number, then √3 = p/q where (p, q) co-prime.

or q = p/√3

or q² = p^2/3 ….(1)

Thus, p² must be divisible by 3. Hence p will also be divisible by 3.

We can write p = 3k, where k is a constant.

⇒ p² = 9c²

(1)⇒

q² = 9c^2/3 = 3c²

or c² = q^2/3

Thus, q² must be divisible by 3, which implies that q will also be divisible by 3.

Thus, both p and q are divisible by 3.

Which is a contradiction, as we assume that p and q are co-prime.

Thus, √3 is irrational.

Hence, the statement p is true.

Let the given statement be false i.e., ~ p ∶ \(\sqrt{11}\)

Is rational. The \(\sqrt{11}\) = \(\frac{p}{q}\) where p and q are coprime and q ≠ 0.

⇒ 11 = \(\frac{p^2}{q^2}\)

⇒ p² = 11q²

⇒ 11 divides p ...(1)

∴ r ∈ z such that−

p = 11r

⇒ p² = 121r

⇒ 11q² = 121r²

⇒ q² = 11r²

⇒ 11 divides q … (2)

From (1) & (2), we arrive at a contradiction, since p and q are coprime.

⇒ \(\sqrt{11}\) is irrational.

(i) For all even integers x, x² is also even.

The quantifier is “For All”.

The negation of the statement is-

“There exists an even integer x such that x² is not even”.

(ii) There exists a number which is a multiple of 6 and 9.

The quantifier is “There exists”.

The negation of the statement is-

“There does not exist a number which is a multiple of both 6 and 9”.

(i) The square of an odd number is odd

If a number is odd, then its square is also odd.

(ii) The unit digit of a number is 0 or 5, if it is divisible by 5.

If a number is divisible by 5, then its unit digit it 0 or 5.

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 complex number are real numbers.”

In the bi conditional statement, we use if and only if.

p: A natural number n is odd.

q: q : Natural number n is not divisible by 2.

Then,

p ↔ q = A natural number is odd if and only if it is not divisible by 2.

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

Contrapositive: If natural number ‘n’ is not divisible by 2 or 3, then n is not divisible by 6.