Chapter 14: Mathematical Reasoning

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.

(ii) For all real numbers x and y, xy = y x.

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.

(iii) There exists a real number which is not a rational number.

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 which is not a rational number.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(iv) For every natural number x, x + 1 is also a natural number.

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.

(v) For all real numbers x with x > 3, x 2 is greater than 9.

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.

(vi) There exists a triangle which is not an isosceles triangle.

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.

(vii) For all negative integers x, x 3 is also a negative integers.

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 negative integers x, x³ is also a negative integers.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

(viii) There exists a statement in above statements which is not true.

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 statement in above statements which is not true.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(ix) There exists a even prime number other than 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 even prime number other than 2.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

(x) There exists a real number x such that x 2 + 1 = 0.

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.

Given n³-n

Let us assume, n is even

Let n = 2k, where k is natural number

n³ – n = (2k)³– (2k)

n³ – n = 2k (4k²-1)

Let k (4k² – 1) = m

n³ – n = 2m

Therefore, (n³-n) is even.

Now, let us assume n is odd

Let n = (2k + 1), where k is natural number

n³ – n = (2k + 1)³ – (2k + 1)

n³ – n = (2k + 1) [(2k + 1)² – 1]

n³ – n = (2k + 1) [(4k² + 4k + 1 – 1)]

n³ – n = (2k + 1) [(4k² + 4k)]

n³ – n = 4k (2k + 1) (k + 1)

n² – n = 2.2k (2k + 1) (k + 1)

Let λ = 2k (2k + 1) (k + 1)

n³ – n = 2λ

Therefore, n³-n is even.

Hence, n³-n is always even

p: 125 is divisible by 5 and 7

Let, q: 125 is divisible by 5.

r: 125 is divisible 7.

Here, q is true and r is false.

Therefore, q ᴧ r is False

Hence, p is not valid.

(ii) q: 131 is a multiple of 3 or 11.

Given q : 131 is a multiple of 3 or 11

Let, P: 131 is a multiple of 3.

Q: 131 is a multiple of 11.

Here, P is false and Q is False

Therefore, P ⋁ Q is False

Hence, q is not valid

Let p is false, as the sum of an irrational number and a rational number is irrational.

Let √λ is irrational and n is rational number

√λ + n = r

√λ = r – n

But, we know that √λ is irrational whereas (r-n) is rational which is contradiction.

Here, Our Assumption is False

Hence, P is true.

Given for any real number x, y if x=y

Now we have to find x² = y²

Let us assume

p: x=y where x and y are real number

On squaring both sides we get

x² = y² : q (Assumption)

Therefore,

p = q

Hence, Proved

Let us assume

p: n² is an even integer.

~p: n is not an even integer

q: n is also an even integer

~q=n is not an even integer.

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

Therefore,

~q → ~p = If n is not an even integer then n² is not an even integer.

Hence, ~q is true → ~p is true.

Objective type questions:

Choose the correct answer out of the four options given against each of the Exercises 17 to 36 (M.C.Q.).

(C) 6 is a natural number.

Explanation:

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

Here, 6 is a natural number is true

18. Which of the following is not a statement

(A) Smoking is injurious to health.

(B) 2 + 2 = 4

(C) 2 is the only even prime number.

(D) Come here.

(D) Come here.

Explanation:

To given order like Come here, Go there are not statements.