Which of the following sentences are statements? In case of a statement mention whether it is true or false.

(i) The sun is a star.

(ii) √7 is an irrational number.

(iii) The sum of 5 and 6 is less than 10.

(iv) Go to your class.

(v) Ice is always cold.

(vi) Have you ever seen the Red Fort?

(vii) Every relation is a function.

(viii) The sum of any two sides of a triangle is always greater than the third side.

(ix) May God bless you!

Which of the following sentences are statements? In case of a statement, mention whether it is true or false.

(i) Paris is in France.

(ii) Each prime number has exactly two factors.

(iii) The equation x² + 5|x| + 6 = 0 has no real roots.

(iv) (2 + √3) is a complex number.

(v) Is 6 a positive integer?

(vi) The product of -3 and -2 is -6.

(vii) The angles opposite the equal sides of an isosceles triangle are equal.

(viii) Oh! It is too hot.

(ix) Monika is a beautiful girl.

(x) Every quadratic equation has at least one real root.

Which of the following statements are true and which are false? In each case give a valid reason for your answer.

(i) p: √11 is an irrational number

(ii) q: Circle is a particular case of an ellipse.

(iii) r: Each radius of a circle is a chord of the circle

(iv) S: The center of a circle bisects each chord of the circle

(v) t: If a and b are integers such that a < b, then –a > -b.

(vi) y: The quadratic equation x² + x + 1 = 0 has no real roots

Write the negation of each of the following statements:

(i) Every natural number is greater than 0.

(ii) Both the diagonals of a rectangle are equal.

(iii) The sum of 4 and 5 is 8.

(iv) The number 6 is greater than 4.

(v) Every natural number is an integer.

(vi) The number -5 is a rational number

(vii) All cats scratch.

(viii) There exists a rational number x such that x² = 3.

(ix) All students study mathematics at the elementary level.

(x) Every student has paid the fees.

(xi) There is some integer k for which 2k = 6.

(xii) None of the students in this class has passed.

Split each of the following into simple sentences and determine whether it is true or false.

(i) A line is straight and extends indefinitely in both the directions.

(ii) A point occupies a position, and its location can be determined.

(iii) The sand heats up quickly in the sun and does not cool down fast at night.

(iv) 32 is divisible by 8 and 12.

(v) x = 1 and x = 2 are the roots of the equation x² – x – 2 = 0.

(vi) 3 is rational, and √3 is irrational.

(vii) All integers are rational numbers, and all rational numbers are not real numbers.

(viii) Lucknow is in Uttar Pradesh, and Kanpur is in Uttarakhand.

Split each of the following into simple sentences and determine whether it is true or false. Also, determine whether an ‘inclusive or’ or ‘exclusive or’ is used.

(i) The sum of 3 and 7 is 10 or 11.

(ii) (1 + i) is a real or a complex number.

(iii) Every quadratic equation has one or two real roots.

(iv) You are wet when it rains, or you are in a river.

(v) 24 is a multiple of 5 or 8.

(vi) Every integer is rational or irrational.

(vii) For getting a driving license, you should have a ration card or a passport.

(viii) 100 is a multiple of 6 or 8.

(ix) Square of an integer is positive or negative.

(x) Sun rises or Moon sets.

Find the truth set in case of each of the following open sentences defined on N:

(i) x + 2 < 10

(ii) x + 5 < 4

(iii) x + 3 > 2

Let A = [2, 3, 5, 7]. Examine whether the statements given below are true or false.

(i) ∃ x ∈ A such that x + 3 > 9.

(ii) ∃ x ∈ A such that x is even.

(iii) ∃ x ∈ A such that x + 2 = 6.

(iv) ∀ x ∈ A, x is prime.

(v) ∀ x ∈ A, x + 2 < 10.

(vi) ∀ x ∈ A, x + 4 ≥ 11

Rewrite the following statement in five different ways conveying the same meaning.

If a given number is a multiple of 6, then it is a multiple of 3.

Write each of the following statements in the form ‘if …. then’ :

(i) A rhombus is a square only if each of its angles measures 90⁰.

(ii) When a number is a multiple of 9, it is necessarily a multiple of 3.

(iii) You get a job implies that your credentials are good.

(iv) Atmospheric humidity increase only if it rains

(v) If a number is not a multiple of 3, then it is not a multiple of 6.

Write the converse and contrapositive of each of the following :

(i) If x is a prime number, then x is odd.

(ii) If a positive integer n is divisible by 9, then the sum of its digits is divisible by 9.

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

(iv) If the diagonal of a quadrilateral bisect each other, then it is a parallelogram.

(v) If A and B are subsets of X such that A ⊆ B, then (X – B) ⊆ (X – A)

(vi) If f(2) = 0, then f(x) is divisible by (x – 2).

(vii) If you were born in India, then you are a citizen of India.

(viii) If it rains, then I stay at home.

Given below are some pairs of statements. Combine each pair using if and only if :

(i) p : If a quadrilateral is equiangular, then it is a rectangle.

q : If a quadrilateral is a rectangle, then it is equiangular.

(ii) p : If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

q : If a number is divisible by 3, then the sum of its digits is divisible by 3.

(iii) p : A quadrilateral is a parallelogram if its diagonals bisect each other.

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

(iv) p : If f(a) = 0, then (x – a) is a factor of polynomial f(x).

q : If (x – a) is a factor of polynomial f(x), then f(a) = 0.

(v) p : If a square matrix A is invertible, then |A| is nonzero.

q : If A is a square matrix such that |A| is nonzero, then A is invertible.

write each of the following using ‘if and only if’ :

(i) In order to get A grade, it is necessary and sufficient that you do all the homework regularly.

(ii) If you watch television, then your mind is free, and if your mind is free, then you watch television.

Let p : If x is an integer and x² is even, then x is even,

Using the method of contrapositive, prove that p is true.

Consider the statement :

q : For any real numbers a and b, a² = b²⇒ a = b

By giving a counter-example, prove that q is false.

By giving a counter-example, show that the statement is false :

p : If n is an odd positive integer, then n is prime.

Use contradiction method to prove that :

is irrational

is a true statement.

By giving a counter-example, show that the following statement is false :

p : If all the sides of a triangle are equal, then the triangle is obtuse angled.