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#### Question 1:

Find the component statements of the following compound statements:
(i) The sky is blue and the grass is green.
(ii) The earth is round or the sun is cold.
(iii) All rational numbers are real and all real numbers are complex.
(iv) 25 is a multiple of 5 and 8.

​(i) The component statements of the given compound statement are:
1) The sky is blue.
2)The grass is green

(ii) The component statements of the given compound statement are:
1)The earth is round.
2)The sun is cold.

(iii) The component statements of the given compound statement are:
1) All rational numbers are real.
2) All real numbers are complex.

(iv) The component statements of the given compound statement are:
1) 25 is a multiple of 5.
2) 25 is a multiple of 8.

#### Question 2:

For each of the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.
(i) Students can take Hindi or Sanskrit as their third language.
(ii) To entry a country, you need a passport or a voter registration card.
(iii) A lady gives birth to a baby boy or a baby girl.
(iv) To apply for a driving licence, you should have a ration card or a passport.

​(i) Exclusive OR is used because students can opt for either Hindi or Sanskrit as their third language.

(ii) Inclusive OR is used because a person can have both passport as well as voter registration card.

(iii) Exclusive OR because a lady can give a birth to a baby who is either a boy or a girl.

(iv) Inclusive OR because a person could have both ration card as well as passport.

#### Question 3:

Write the component statements of the following compound statements and check whether the compound statement is true or false:
(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.
(ii) All rational numbers are real and all real numbers are not complex.
(iii) Square of an integer is positive or negative.
(iv) x = 2 and x = 3 are the roots or the equation 3x2x − 10 = 0.
(v) The sand heats up quickly in the sun and does not cool down fast at night.

​(i) The component statements of the given compound statement are:
1) To enter into a public library, children need an identity card from the school.
2) To enter into a public library, children need a letter from the school authorities.
The compound statement is true because both component statements are true.

(ii) The component statements of the given compound statement are:
1) All rational numbers are real.
2) All real numbers are not complex.
The compound statement is false because all real numbers are complex. The connective used is "and". So, even if one component statement is false, the compound statement is false.

(iii) The component statements of the given compound statement are:
1) Square of an integer is positive.
2) Square of an integer is negative.
The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.

(iv) The component statements of the given compound statement are:
1) $x=2$ is the root or the equation $3{x}^{2}-x-10=0$.
2) $x=3$
is the root or the equation  $3{x}^{2}-x-10=0$.
The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement "$x=3$ is the root or the equation $3{x}^{2}-x-10=0$"  is false. Therefore, the compound statement is false.

(v) The component statements of the given compound statement are:
1) The sand heats up quickly in the sun.
2) Sand does not cool down fast at night.
The compound statement uses "and" as the connective. For the compound statement to be true, both the component statements must be true. The second component statement "Sand does not cool down fast at night" is false. Sand cools down fast at night. Therefore, the compound statement is false.

#### Question 4:

Determine whether the following compound statements are true or false:
(i) Delhi is in India and 2 + 2 = 4.
(ii) Delhi is in England and 2 + 2 = 4.
(iii) Delhi is in India and 2 + 2 = 5.
(iv) Delhi is in England and 2 + 2 =5.

(i) True
Both component statements are true and the connective is "and".
(ii) False
The first component statement "Delhi is in England" is false. Since the connective is "and" and one component statement is false, the compound statement is false.
(iii) False
The second component statement "2 plus 2 equals 5" is false. Since the connective is "and" and one component statement is false, the compound statement is false.
(iv) False
Both the component statements are false. Therefore, the compound statement is false.

#### Question 1:

Write the negation of each of the following statements:
(i) For every x ϵ N, x + 3 < 10
(ii) There exists x ϵ N, x + 3 = 10

(i) Negation of the given statement:
There exists a number x such that $x+3\ge 10$.

(ii) Negation of the given statement:
For every x ϵ N, $x+3\ne 10$.

#### Question 2:

Negate each of the following statements:
(i) All the students completed their homework.
(ii) There exists a number which is equal to its square.

(i) Negation of the given statement:
Some students did not complete their homework.

(ii) Negation of the given statement:
There exists a number which is not equal to its square.

#### Question 1:

Write each of the following statements in the form "if p, then q".
(i) You can access the website only if you pay a subscription fee.
(ii) There is traffic jam whenever it rains.
(iii) It is necessary to have a passport to log on to the server.
(iv) It is necessary to be rich in order to be happy.
(v) The game is cancelled only if it is raining.
(vi) It rains only if it is cold.
(vii) Whenever it rains it is cold.
(viii) It never rains when it is cold.

(i) If you pay a subscription fee, then you can access the website.
(ii) If it rains, then there is a traffic jam.
(iii) If you want to log on to the server, then you need a passport.
(iv) If you want to be happy, then you will have to be rich.
(v) If it rains, only then the game is cancelled.
(vi) If it rains, then it is cold.
(vii) If it rains, then it is cold.
(viii) If it is cold, then it never rains.

#### Question 2:

State the converse and contrapositive of each of the following statements:
(i) If it is hot outside, then you feel thirsty.
(ii) I go to a beach whenever it is a sunny day.
(iii) A positive integer is prime only if it has no divisors other than 1 and itself.
(iv) If you live in Delhi, then you have winter clothes.
(v) If a quadrilateral is a parallelogram, then its diagonals bisect each other.

(i) Converse of the given statement:
If you feel thirsty, then it is hot outside.
Contrapositive of the given statement:
If you do not feel thirsty, then it is not hot outside.

(ii) Converse of the given statement:
If I go to a beach, then it is a sunny day.
Contrapositive of the given statement:
If I do not go to a beach, then it is not a sunny day.

(iii) Converse of the given statement:
If a positive integer has no divisors other than 1 and itself, then it is prime.
Contrapositive of the given statement:
If a positive integer has some divisors other than 1 and itself, then it is not prime.

(iv) Converse of the given statement:
If you have winter clothes, then you live in Delhi.
Contrapositive of the given statement:
​If you do not have winter clothes, then you do not live in Delhi.

(v) Converse of the given statement:
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Contrapositive of the given statement:
If the diagonals of a quadrilateral do not bisect each other, then it is not a parallelogram.

#### Question 3:

Rewrite each of the following statements in the form "p if and only if q".
(i) p : If you watch television, then your mind is free and if your mind is free, then you watch television.
(ii) q : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.
(iii) r : For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.
(iv) s : If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

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

(ii) A quadrilateral is a rectangle if and only if it is equiangular.

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

(iv) The tumbler is half empty if and only if the tumbler is half full.

#### Question 4:

Determine the contrapositive of each of the following statements:
(i) If Mohan is a poet, then he is poor.
(ii) Only if Max studies will he pass the test.
(iii) If she works, she will earn money.
(iv) If it snows, then they do not drive the car.
(v) It never rains when it is cold.
(vi) If Ravish skis, then it snowed.
(vii) If x is less than zero, then x is not positive.
(viii) If he has courage he will win.
(ix) It is necessary to be strong in order to be a sailor.
(x) Only if he does not tire will he win.
(xi) If x is an integer and x2 is odd, then x is odd.

(i) If Mohan is not poor, then he is not a poet.
(ii) If Max does not study, then he will not pass the test.
(iii) If she does not earn money, then she will not work.
(iv) If they do not drive the car, then there is no snow.
(v)  If it rains, it is not cold.
(vi) If it did not snow, then Ravish does not ski.
(vii) If x is positive, then x is not less than zero.
(viii) If he does not win, then he does not have courage.
(ix)  If you are not strong, then you cannot be a sailor.
(x)  If he tires, then he will not win.
(xi) If x is even, then x2 is even.

#### Question 1:

Check the validity of the following statements:
(i) p : 100 is a multiple of 4 and 5.
(ii) q : 125 is a multiple of 5 and 7.
(iii) r : 60 is a multiple of 3 or 5.

(i) p: 100 is a multiple of 4 and 5.
Since 100 is a multiple of 4 and 5, the statement is true.
Hence, it is a valid statement.

(ii) q: 125 is a multiple of 5 and 7.
Since 125 is a multiple of 5 but is not a multiple of 7, the statement is not true.
Hence, it is not a valid statement.

(iii) r: 60 is a multiple of 3 or 5.
Since 60 is a multiple of 3 and 5, the statement is true.
Hence, it is a valid statement.

#### Question 2:

Check whether the following statement are true or not:
(i) p : If x and y are odd integers, then x + y is an even integer.
(ii) q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

(i) p: If x and y are odd integers, then x + y is an even integer.
Let q and r be two statements.
Here,
q: x and y are odd integers.
r: x + y is an even integer.
Let q be true.
Then, q is true.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$⇒$x + y = (2m + 1) + (2n + 1)
$⇒$x + y = 2m + 2n + 2 = 2(m + n + 1)
So, x + y is an even integer.
Hence, the statement is true.

ii)
p: If x and y are integers such that xy is even, then at least one of x and y is an even integer.
Let q and r be two statements.
Here,
q: xy is an even integer.
r: At least one of x and y is an even integer.
Let r be not true.
Then, r is not true.
It is false that at least one of x and y is an even integer.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$⇒$xy =(2m + 1)(2n + 1)
$⇒$xy = 2(2mn + m + n) + 1
So, xy is not an even integer. Thus, xy is not true.
Hence, the statement is true.

#### Question 3:

Show that the statement
p : "If x is a real number such that x3 + x = 0, then x is 0"
is true by
(i) direct method
(ii) method of contrapositive

p : "If x is a real number such thatthen x is 0".
Let q and r be the statements.
Here,
q: x is a real number such that .

r: x is 0.

(i) Direct method
Let q be true.
To obtain we have:

or, x = 0
Thus, r is true.
Hence, "if q, then r" is a true statement.

(ii) Method of contrapositive
Let r not be true.
r is not 0.

If , then q is not true.
Hence, "if ~q, then ~r" is a true statement.

Let q not be true.
Then,
$~$q is true
$~$($⇒$r) is true.

q &$~$is true

x is a real number such that ${x}^{3}+x=0$

Then, x is not 0.

x = 0 and x$\ne$0
Hence, q is true.

#### Question 4:

Show that the following statement is true by the method of contrapositive
p : "If x is an integer and x2 is odd, then x is also odd"

Let and be statements.
Here,
q: If x is an integer and x2 is odd.
r
: x is also odd.
Then, p is" if q, then r".
Let r be false.
So, x is not an odd integer, i.e., x is an even integer.
Let:
for some integer n
or, ${x}^{2}=4{n}^{2}$
In other words, x2 is an even integer.
Now, q is false.
Thus, r is false; this implies q is false.
Hence, p is a true statement.

#### Question 5:

Show that the following statement is true
"The integer n is even if an only if n2 is even"

The given statement can be rewritten as:
"The necessary and sufficient condition for integer n to be even is n2 must be even".

Let and q be the following statements.
p: The integer n is even.
q: n​2 is even.
The given statement is "p if and only if q".
​To check its validity, we have to check the validity of the following statements:
(i) If p, then q.
(ii) If q, then p.
Checking the validity of "if p, then q"
"
If the integer n is even, then n​2 is even."
Let us assume that is even.
Then, $n=2m$, where m is an integer.
Thus, we have:
${n}^{2}=4{m}^{2}$
Here, nis even.
Therefore, "if p, then q" is true.
The statement "if q, then p" is given by
"If n is an integer and n2 is even, thenis even".
To check he validity of the statement, we will use the contrapositive method. So, letbe an integer. Then,
is odd.
Here, $n=2k+1$ for some integer k.
$⇒$${n}^{2}=4{k}^{2}+2k+1$
Then, n​2 is an odd integer.
n2
is not an even integer.
Thus "if q, then p" and "p if and only if q" are true.

#### Question 6:

By giving a counter example, show that the following statement is not true.
p : "If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle".

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.

Statement p has to be proved false.
For this purpose, we need to prove that if q, then ~r.

To show this, none of the angles of the triangle should be obtuse.

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

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

Hence, it can be concluded that statement p is false.

#### Question 7:

Which of the following statements are true and which are false? In each case give a valid reason for saying so
(i) p : Each radius of a circle is a chord of the circle.
(ii) q : The centre of a circle bisects each chord of the circle.
(iii) r : Circle is a particular case of an ellipse.
(iv) s : If x and y are integers such that x > y, then − x < − y.
(v) t : $\sqrt{11}$ is a rational number.

(i) The given statement is false.

According to the definition of a chord, it should intersect the circumference of a circle at two distinct points.

(ii) The given statement is false.

If a chord is not the diameter of a circle, then the centre does not bisect that chord. In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.

(iii) Equation of an ellipse:

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

If we put a = b = 1, then we obtain ${x}^{2}+{y}^{2}=1$, which is an equation of a circle. Therefore, a circle is a particular case of an ellipse.

Thus, the statement is true.

(iv) y

⇒ –x < –y     (By the rule of inequality)

Thus, the given statement is true.

(v) 11 is a prime number and we know that the square root of any prime number is an irrational number. Therefore, $\sqrt{11}$ is an irrational number.

Thus, the given statement is false.

#### Question 8:

Determine whether the argument used to check the validity of the following statement is correct:
p : "If x2 is irrational, then x is rational"
The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.

The argument used to check the validity of the given statement is false because it is given in the argument that x2 = π 2 is irrational and therefore x = π is irrational; however, according to the statement, x is rational.

#### Question 1:

Find out which of the following sentences are statements and which are not. Justify your answer.
(i) Listen to me, Ravi !
(ii) Every set is a finite set.
(iii) Two non-empty sets have always a non-empty intersection.
(iv) The cat pussy is black.
(v) Are all circles round?
(vi) All triangles have three sides.
(vii) Every rhombus is a square.
(viii) x2 + 5 | x | + 6 = 0 has no real roots.
(ix) This sentence is a statement.
(x) Is the earth round?
(xi) Go !
(xii) The real number x is less than 2.
(xiii) There are 35 days in a month.
(xiv) Mathematics is difficult.
(xv) All real numbers are complex numbers.
(xvi) The product of (−1) and 8 is 8.

(i) Listen to me, Ravi!
It is an exclamatory sentence. Therefore, it is not a statement.

(ii) Every set is a finite set.
It is a false assertive sentence because there are some sets that are infinite like
the set of all real numbers. Therefore, it is a statement.

(iii) Two non-empty sets have always a non-empty intersection.
It is a false assertive sentence. Two non-empty sets with no common elements can have an empty intersection. Therefore, it is a statement.

(iv) The pussy cat is black.
It is a declarative sentence, which may be true or false but cannot be both at the same time, so it is a statement.

(v) Are all circles round?
It is an interrogative sentence, so it is not a statement.

(vi) All triangles have three sides.
It is a true declarative sentence because a figure that has three sides is a triangle. Thus, it is a true statement.

(vii) Every rhombus is a square.
It is not true that every rhombus is a square because some rhombi may have all angles other than 90. So, it is a false statement.

(viii)
x2 + 5 | x | + 6 = 0 has no real roots.
It is a true declarative sentence, so it is a statement.

(ix) This sentence is a statement.
Without knowing the sentence, we cannot decide whether it is true or false. So, it is not a statement.

(x) Is the earth round?
It is an interrogative sentence, so it is not a statement.

(xi) Go!
It is an exclamatory sentence, so it is not a statement.

(xii) The real number
x is less than 2.
We cannot decide whether this sentence is true or false without knowing the value of $x$. So, it is not a statement.

(xiii) There are 35 days in a month.
It is a false assertive sentence, so it is a false statement.

(xiv) Mathematics is difficult.
Mathematics could be easy for some people, so this sentence may or may not be true. So, it is not a statement.

(xv) All real numbers are complex numbers.
It is true because we can write a real number as $x+0i$. So, it is a true statement.

(xvi) The product of (−1) and 8 is 8.

It is an assertive sentence; therefore, it is a statement. But $-1×8=-8$; therefore, the statement is false.

#### Question 2:

Give three examples of sentences which are not statements. Give reasons for the answers.

1) I won the trophy!
It is an exclamatory sentence, so it is not a statement.

2) Please fetch me a glass of water.
It is an imperative sentence. In other words, it can be expressed either as a request or as a command. Therefore, it not a statement.

3) Can you do this work for me?
It is an interrogative sentence, so it is not a statement.

#### Question 1:

Write the negation of the following statements:
(i) Banglore is the capital of Karnataka.
(ii) It rained on July 4, 2005.
(iii) Ravish is honest.
(iv) The earth is round.
(v) The sun is cold.

(i) Negation of the given statement:
It is not true that Bangalore is the capital of Karnataka.
Or
Bangalore is not the capital of Karnataka.

(ii) Negation of the given statement:
It is not true that it rained on July 4, 2005.
Or
It did not rain on July 4, 2005.

(iii) Negation of the given statement:
It is not true that Ravish is honest.
Or
Ravish is not honest.

(iv) Negation of the given statement:
The earth is not round.
Or
It is not true that the earth is round.

(v) Negation of the given statement:
The sun is not cold.
Or
It is not true that the sun is cold.

#### Question 2:

(i) All birds sing.
(ii) Some even integers are prime.
(iii) There is a complex number which is not a real number.
(iv) I will not go to school.
(v) Both the diagonals of a rectangle have the same length.
(vi) All policemen are thieves.

(i) Negation of the given statement:
Some birds do not sing.
Or
There exists a bird that does not sing.

(ii) Negation of the given statement:
Some integers are not prime.
Or
No even integer is prime.

(iii) Negation of the given statement:
All complex numbers are real numbers.

(iv) Negation of the given statement:
I will go to school.

(v) Negation of the given statement:
Both the diagonals of a rectangle do not have the same length.
Or
Both the diagonals of a rectangle have different lengths.

(vi) Negation of the given statement:
There exists a policeman who is not a thief.
Or
At least one policeman is not a thief.

#### Question 3:

Are the following pairs of statements are negation of each other:
(i) The number x is not a rational number.
The number x is not an irrational number.
(ii) The number x is not a rational number.
The number x is an irrational number.

​(i) The number x is not a rational number.
The number x is not an irrational number.
The statements in this pair are the negation of each other.

(ii) The number x is not a rational number.
The number x is an irrational number.​
The statements in this pair are not the negation of each other because both statements are the same. Both the statements convey that x is an irrational number.

#### Question 4:

Write the negation of the following statements:
(i) p : For every positive real number x, the number (x − 1) is also positive.
(ii) q : For every real number x, either x > 1 or x < 1.
(iii) r : There exists a number x such that 0 < x < 1.

(i) p: For every positive real number x, the number (x − 1) is also positive.
~p: At least for one positive real number x, the number $\left(x-1\right)$ is not positive.

(ii)
q: For every real number x, either x > 1 or x < 1.
~q: At least for one real number x, neither nor .

(iii)
r: There exists a number x such that 0 < x < 1.
~r: For every real number x, either or
.

#### Question 5:

Check whether the following pair of statements are negation of each other. Give reasons for your answer.
(i) a + b = b + a is true for every real number a and b.
(ii) There exist real numbers a and b for which a + b = b + a.

​The given statements are not negation of each other because the negation of "$a+b=b+a$ is true for every real number a and b" is "There exist real numbers a and b for which $a+b\ne b+a$".