Mathematical Reasoning

(i) Since there are 28 or 30 or 31 days in a month.

It is a false Sentence

Hence it is a statement.

(ii) Mathematics is difficult for some but not for other.

Hence this statement can be both true or false.

So it is not a statement.

(iii) 5+7 = 12 which is greater than 10

It is true

Hence it is a statement.

(iv) This sentence may be true or may not be.

For example 2² = 4(even)

3² = 9(odd)

Hence the square of a natural number may be even or may be odd.

Hence this is not a statement.

(v) Square is quadrilateral & has equal length (all four sides are equal). But, trapezium is quadrilateral which has unequal length.

(Only two sides are equal)

Thus, Sentence is sometimes true & sometimes false.

Hence it is not a statement.

(vi) Since it is order.

Hence it is not a statement.

(vii) It is false sentence.

Because product of (-1) & 8 is -8.

Hence it is a statement.

(viii) This Sentence is always true.

Because the sum of interior angles of a triangle is 180°.

Hence it is a statement.

(ix) It is not clear which day is windy day.

So, it may be true or maybe false.

Hence it is not a statement.

(x) All real numbers can be written in the form of

a + i0 (when 'a' is real number)

So, all real numbers are complex numbers

Hence, it is always true.

Hence it is a statement.

Example 1

Mathematics is easy

Since it is sometimes true and sometimes false.

It is not a statement.

Example 2

How old are you?

Since it is a question.

It is not statement as statement are either true or false sentence.

Example 3

What a match!

Since it is an exclamation.

It is not statement as statement are either true or false sentence.

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

Or

It is false to say that Chennai is the capital of Tamil Nadu.

Or

It is not the case that Chennai is the capital of Tamil Nadu.

(ii) √2 is a complex number.

Or

It is false to say that √2 is not a complex number.

Or

It is not the case that √2 is not a complex number.

(iii) All triangles are equilateral triangle.

Or

It is false to say that all triangles are not equilateral triangle.

Or

It is not the case that all triangles are not equilateral triangle.

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

Or

It is false to say that the number 2 is greater than 7.

Or

It is not the case that the number 2 is greater than 7.

(v) Every natural number is not an integer.

Or

It is false to say that Every natural number is an integer.

Or

It is not the case that Every natural number is an integer.

(i) The negation of the first statement is

The number x is a rational number.

Which is same as second statement

The number x is not an irrational number.

Hence the given pairs are negation of each other.

(ii) The negation of the first statement is

The number x is not a rational number.

Which is same as second statement

The number x is an irrational number.

Hence the given pairs are negation of each other.

(i) The component statements are:-

p: Number 3 is prime.

q: It is odd.

The connecting word is 'Or'

We know that number 3 is prime

Hence p is true

We know that number 3 is odd

Hence q is true.

(ii) The component statements are:-

p: All integers are positive.

q: All integers are negative.

The connecting word is 'Or'

Since all integers are not positive

Hence p is false

Since all integers are not negative

Hence q is false.

(iii) The component statements are:-

p: 100 is divisible by 3.

q: 100 is divisible by 11.

r: 100 is divisible by 5.

The connecting word is 'and'

Since, p is false, because 100 is not divisible by 3

q is false, because 100 is not divisible by 11

r is true, because 100 is divisible by 5

Hence p & q are false & r is true.

(i) Here, the connecting word is 'and'.

The component statements are

p: All rational numbers are real.

q: All real numbers are not complex.

(ii) Here, the connecting word is 'Or'.

The component statements are

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

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

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

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

(i) Quantifier is "There exists".

Negation of statement is

There does not exist a number which is equal to its square.

(ii) Quantifier is "For Every".

Negation of statement is

There exist a real number x such that x is not less than x + 1.

(iii) Quantifier is "There exists".

Negation of statement is

There does not exist a capital for every state in India.

The negation of first statement is

"There exists no real numbers x and y for which x + y = y + x"

Which is not equal to second statement.

Hence given statement is false.

(i) We know that

Sun rises in the morning. At the same time, moon sets

Hence both can happen simultaneously

Hence it is inclusive.

(ii) A person can have ration card or a passport or both to apply for a driving license.

Hence, here 'Or' is inclusive.

(iii) All integers are either positive or negative but cannot be both.

Hence, here 'Or' is exclusive.

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 the 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.

Contrapositive - It is done by Adding 'not' to both component statements & charging order

p ⇒ q

~q ⇒ ~p

Converse - It is done by changing order of statement

p ⇒ q

then q ⇒ p

(i) Contrapositive

If a number x is not odd, then it is not prime number.

Converse

If a number x is odd, then x is a prime number.

(ii) Contrapositive

If two lines intersect in the same plane, then they are not parallel.

Converse

If two lines do not intersect in the same plane, then they are parallel.

(iii) Contrapositive

If something is not at low temperature, then it is not cold.

Converse

If something is at low temperature, then it is cold.

(iv) Contrapositive

If you know how to reason deductively, then you can comprehend geometry.

Converse

If you do not know how to reason deductively, then you can not comprehend geometry.

(v) This statement is not in if-then form

Writing in if-then form

This statement can be written as "if x is an even number, then x is divisible by 4".

Contrapositive

If x is not divisible by 4, then x is not an even number.

Converse

If x is divisible by 4, then x is an even number.

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

(ii) If the banana tree stays warm for a month, then it will bloom.

(iii) If diagonals of quadrilateral bisect each other, then it is a parallelogram.

(iv) If you get A+ in the class, then you have done all the exercises in the book.

(a) The given statement is "if p then q"

Contrapositive is "~q, then ~p"

So, (i) is Contrapositive statement.

Converse is "q, then p"

So, (ii) is converse statement.

(b) The given statement is "if p then q"

Contrapositive is "~q, then ~p"

So, (i) is Contrapositive statement.

Converse is "q, then p"

So, (ii) is converse statement.

(i) The given statement is of the form if p then q

Here

p: x is aa real number such that x³ + 4x = 0.

q: x is 0

In Direct Method,

we assume p is true, and prove q is true

So,

Let x be a real number such that x³ + 4x = 0, prove that x = 0

p is true

Consider

x³ + 4x = 0 where x is real

⇒ x(x² + 4) = 0

⇒ x = 0 or x² + 4 = 0

⇒ x = 0 or x² = – 4

⇒ x = 0 or x =

x = is not possible because it is given that x is real.

Hence x = 0 only

⇒ q is true

Hence proved

(ii) p: If x is a real number such that x³ + 4x = 0, then x is 0

Let us assume

x³ + 4x = 0

but x ≠ 0

Solving x³ + 4x = 0 where x is real

⇒ x(x² + 4) = 0

⇒ x = 0 or x² + 4 = 0

⇒ x = 0 or x² = – 4

⇒ x = 0 or x =

x = is not possible because it is given that x is real number

Hence, only solution is x = 0 but we take x ≠ 0

Hence we get a contradiction

Hence our assumption is wrong

Hence x is real number such that x³ + 4x = 0 then x is 0.

(iii) Let p: if x is real number such that x³ + 4x = 0

q: x is 0

The above statement is of the form if p then q

By method of Contrapositive

By assuming that q is false, prove that p must be false

Let q is false

i.e. x is not equal to 0

i.e. x ≠ 0

i.e. x × (positive number) ≠ 0 × (positive number)

i.e. x × (x² + 4) ≠ 0 × (x² + 4)

i.e. x³ + 4x ≠ 0

i.e. p is false

Hence x is real number such that x³ + 4x = 0 then x is 0.

Let a = – 1 & b = 1

Now,

a² = (– 1)² = 1

b² = (1)² = 1

Since a² = b² = 1

But a ≠ b

⇒ – 1 ≠ 1

Hence the given statement is not true.

p: if x is an integer and x² is even, then x is also even.

Let p: if x is an integer and x² is even

q: x is even

The given statement is if p then q

Method of Contrapositive

By assuming q is not true & prove that p must be true

i.e. ~q ⇒ ~p

Let q is not true & prove p is also not true.

⇒ q is not true

i.e. x is not even

i.e. x is odd

i.e. x = 2n + 1

Squaring both side

(x)² = (2n + 1)²

⇒ x² = 4n² + 4n + 1

⇒ x² = 4(n² + n) + 1

⇒ x² is odd

⇒ p is also not true

Hence the given statement is true.

(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 given statement p is false.

(ii) Putting x = 1 in equation

x² – 1 = (1)² – 1 = 1 – 1 = 0

Hence x = 1 be the root of x² – 1 = 0

& x is lying between 0 & 2

Hence the given statement is not true.

(i)

This statement is false

A chord intersect the circle in two points.

OB & OC are radius of circle

But they don't intersect the circle at two points

Hence OB & OC are not the Chord of a circle.

(ii)

The given statement is false

Here AB is a chord as it intersect the circle at two points,

But it does not pass through center O.

So, center of circle does not bisect each chord of the circle.

(iii) This statement is true

Equation of ellipse is

putting a = b

⇒ x² + y² = b²

which is the equation of circle.

So, circle is the particular case of ellipse.

(iv) x>y

Multiplying – 1 both sides

(– 1)x<(– 1)y

⇒ – x < – y

If – 1 is multiplied to both L.H.S & R.H.S, sign of inequality changes.

This is the rule of inequality

Hence the given statement is true.

(v) cannot be written in the form of p/q

Hence is irrational.

Hence the given statement is false.

(i) The negation of the statement p is

It is false that for every positive real number x, the number x – 1 is also positive.

This can be rewritten as

There exists a positive real number x such that the number x – 1 is not positive.

(ii) The negation of the statement q is

It is false that all cats scratch.

This can be rewritten as

There is at least one cat that does not scratch.

(iii) The negation of the statement r is

It is false that for every real number x, either x > 1 or x < 1.

This can be rewritten as

There exists a real number x such that neither x > 1 nor x < 1.

(iv)The negation of the statement s is

It is false that there exists a number x such that 0 < x < 1.

This can be rewritten as

There does not exist a number x such that 0 < x < 1.

We know that

the converse of a given statement “if p, then q” is if q, then p and the contra positive of the statement if p, then q is “if∼q,then∼p”.

(i) The statement p can be rewritten as

If a positive integer is prime, then it has n divisors other than 1 and the number itself.

Converse of statement p is

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

Contra positive of statement p is

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

(ii) The statement q can be rewritten as

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

Converse of statement q is

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

Contra positive of statement q is

If don’t go to a beach, it is not a sunny day.

(iii) Converse of statement r is

If you feel thirsty, then it is hot outside.

Contra positive of statement r is

If you don’t feel thirsty, then it is not hot outside.

(i) The statement p can be written as

If you log on to the server, then you have a password.

(ii) The statement q can be written as

If it rains, then it is a traffic jam.

(iii) The statement r can be written as

If you can access the website, then you pay a subscription fee.

(i) The statement p can be written as

You watch television if and only if your mind is free.

(ii) The statement q can be written as

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

(iii) The statement r can be written as

A quadrilateral is equiangular if and only if it is a rectangle.

The compound statement connecting with “And” is

25 is a multiple of 5 and 8.

The above compound statement is false because 25 is a multiple of 5 but not a multiple of 8.

The compound statement connection with “Or” is

25 is a multiple of 5 or 8.

The above compound statement is true because 25 is not a multiple of 8 but is a multiple of 5.

(i) Assume that the given statement p is false.

So, the statement becomes the sum of an irrational number and a rational number is rational.

Let us take for example,

Where √p is irrational number and q/r and s/t are rational numbers.

Then, is a rational number and √p is an irrational number.

This is a contradiction.

∴The assumption we made is wrong.

Thus, the given statement p is true.

(ii) Assume that the given statement q is false.

So, the statement becomes if n is a real number with n > 3, then n² < 9.

From the given statement, we know that n > 3 and n is a real number.

Squaring on both sides, we get

⇒ n² > 3²

⇒ n² > 9

This is a contradiction.

∴The assumption we made is wrong.

Thus, the given statement q is true.

The statement p in five different ways can be written as follows:

(i) A triangle is equiangular implies it is an obtuse angled triangle.

(ii) Knowing that a triangle as equiangular is sufficient to conclude that it is an obtuse angled triangle.

(iii) A triangle is equiangular only if it is an obtuse angled triangle.

(iv)When a triangle is equiangular, it is necessarily an obtuse angled triangle.

(v) If a triangle is not an obtuse angled triangle, it is not equiangular.