Mathematical Reasoning

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.

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.

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.

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.

Here, The given sentence is an interrogative sentence.

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.

The given sentence is a true declarative sentence.

Hence, it is a true 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 Rhombuses are not a square.

Hence, it is a statement.

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

Here, If x>0,

x²+5|x|+6=0

x²+5x+6=0

x = -3 or x = -2

But, Since x>0, the equation has no roots.

If x<0,

x²+5|x|+6=0

x² - 5x+6=0

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

Hence,it is a statement.

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

Here, We cannot find the truth values of this sentence, because either value contradict the sense of the sentence.

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.

The given sentence is an interrogative sentence.

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.

The given sentence “Go!” is an exclamatory sentence.

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.

It is not a statement because its truth and false value cannot be determined without knowing the value of x.

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.

The sentence “There are 35 days in a month” is a false declarative statement, so it is a false statement.

Hence, It is a statement.

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 people, and it is not true for some people, 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.

The given sentence is always true.

Hence, It is a statement.

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

The given sentence is always false because,

(-1) × (8) = -8

Hence, It is a statement.

(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.

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 “Banglore is not the capital of Karnataka.”

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 “It didn’t rain on July 4, 2005”.

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 “Ravish is not honest.”

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.”

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.”

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 “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.

The negation of the statement is “All complex number are real numbers.”

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.”

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 “There is at least one rectangle whose both diagonals do not have the same length.”

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.

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:

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.

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:

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

is

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

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 : a + b = b + a is a true for every real number a and b.

is

~p : There exist real numbers are a and b for which a+b≠b+a.

So, the given statement is not the negation of the first statement.

(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) 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.

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.

(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.

(i) 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

For every x ∈ N, x + 3 > 10

is

There exist x ∈ N, such that x + 3 ≥ 10.

(ii) 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

For every x ∈ N, x + 3 = 10

is

There exist x ∈ N, such that x + 3 ≠ 10.

(i) 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

All the students complete their homework.

is

Some of the students did not complete their homework.

(ii) The negation of the statement

There exists a number which is equal to its square.

is

For every real number x, x²≠x.

(i) If you access the website, then you pay a subscription fee.

(ii) If it rains, then there is a traffic jam.

(iii) If you log on the server, then you must have a passport.

(iv) If he is happy, then he is rich.

(v) If it is raining, then the game is canceled.

(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.

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.

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 an integer has no divisor other that 1 and itself, then it is prime.

Contrapositive: If an integer has some divisor other than 1 and itself, then it is prime.

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 have winter clothes, then you live in Delhi.

Contrapositive: If you do not have winter clothes, then you do not live in Delhi.

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 the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Contrapositive: If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.

(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) A tumbler is half empty if and only if it is half full.

(i) Statement: If Mohan is a poet, then he is poor.

Contrapositive : If Mohan is not poor, then he is not a poet.

(ii) Statement: Only if Max studies will he pass the test.

Contrapositive : If Max does not study, then he will not pass the test.

(iii) Statement: If she works, she will earn money.

Contrapositive : If she does not earn money, then she does not work.

(iv) Statement: If it snows, then they do not drive the car.

Contrapositive : If then they do not drive the car, then there is no snow.

(v) Statement: It never rains when it is cold.

Contrapositive : If it rains, then it is not cold.

(vi) Statement: If Ravish skis, then it snowed.

Contrapositive : If it did not snow, then Ravish will not ski.

(vii) Statement: If x is less than zero, then x is not positive.

Contrapositive : If x is positive, then x is not less than zero.

(viii) Statement: If he has courage he will win.

Contrapositive : If he does not win, then he does not have courage.

(ix) Statement: It is necessary to be strong in order to be a sailor.

Contrapositive :If he is not strong, then he is not a sailor

(x) Statement: Only if he does not tire will he win.

Contrapositive : If he tries, then he will not win.

(xi) Statement: If x is an integer and x² is odd, then x is odd.

Contrapositive : If x is even, then x² is even.

(i) Statement: 100 is a multiple of 4 and 5.

Here, we know that 100 is a multiple of 4 as well as 5. So, statement p is true.

Hence, The statement is true, Therefore The statement “p” is a valid statement.

(ii) Statement: 125 is a multiple of 5 and 7

Since 125 is a multiple of 5, but it is not a multiple of 7. So, The statement “q” is not a true statement.

Hence, The statement “q” id not a valid statement.

(iii) Statement: 60 is a multiple of 3 or 5.

Here, we know that 60 is a multiple of 3 as well as 5. So, statement r is true.

Hence, The statement is true. Therefore The statement “r” is a valid statement.

Let us Assume that p and q be the statements given by

p: x and y are odd integers.

q: x + y is an even integer

since the given statement can be written as :

if p, then q.

Let p be true . then,

x and y are odd integers

x = 2m+1, y = 2n+1 for some integers m,n

x+y = (2m+1)+(2n+1)

x+y = (2m+2n+2)

x+y = 2(m+n+1)

x + y is an integer

q is true.

Therefore, p is true q is true

Hence, “if p, then q “ is a true statement.”

Let us Assume that p and q be the statements given by

p: x and y are integers and xy is an even integer.

q: At least one of x and y is even.

Let p be true, then

xy is an even integer. So,

xy = 2(n + 1)

Now,

Let x = 2(k + 1)

Now as x is an even integer, xy = 2(k + 1). y is also an even integer.

Now take x = 2(k + 1) and y = 2(m + 1)

xy = 2(k + 1).2(m + 1) = 2.2(k + 1)(m + 1)

Therefore, it is also true.

Hence, the statement is true.

(i) Direct Method:

Let us Assume that q and r be the statements given by

q: x is a real number such that x³+x=0.

r: x is 0.

since, the given statement can be written as :

if q, then r.

Let q be true . then,

x is a real number suc that x³+x = 0

x is a real number such that x(x²+1) = 0

x = 0

r is true

Thus, q is true

Therefore, q is true r is true

Hence, p is true.

(ii). Method of contrapositive

Let r be not true. then,

R is not true

x ≠ 0, x∈R

x(x²+1)≠0, x∈R

q is not true

Thus, -r = -q

Hence, p : q r is true

(iii) Method of Contradiction

If possible, let p be not true. Then,

P is not true

-p is true

-p(pr) is true

q and –r is true

x is a real number such that x³+x = 0and x≠ 0

x =0 and x≠0

This is a contradiction

Hence, p is true

Let us Assume that q and r be the statements given

q: x is an integer and x² is odd.

r: x is an odd integer.

since the given statement can be written as :

p: if q, then r.

Let r be false . then,

x is not an odd integer, then

x is an even integer

x = (2n) for some integer n

x² = 4n²

x² is an even integer

Thus, q is False

Therefore, r is false q is false

Hence, p: “ if q, then r” is a true statement.

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

Therefore, n² is an even number.

So, q is true when p is true.

Hence, the statement is true.

let us consider a triangle ABC with all angles equal,

Then, each angle of the triangle is equal to 60.

So, ABC is not an obtuse angle triangle.

Therefore, The statement “p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle” is False.

(i) This statement is False

Because the Radius of the circle is not it chord

(ii) This statement is False

Because A chord does not have to pass through the center.

(iii) This statement is true,

Because a circle can be an ellipse in a particular case when the circle has equal axes.

(iv) This statement is true,

Because for any two integer, if x – y id positive then –(x-y) is negative

(v) This statement is False

Because square root of prime numbers are irrational numbers.

Argument Used: x² = π² is irrational, therefore x = π is irrational.

p: “If x² is irrational, then x is rational.”

Let us take an irrational number given by x = √k, where k is a rational number.

Squaring both sides, we get,

x² = k

Therefore, x² is a rational number and contradicts our statement.

Hence, the given argument is wrong.