Identify the quantifiers and write the negation of the following statements
There exists a number which is a multiple of 6 and 9

Which of the following sentences are statements? Give reasons for your answer.
(i)There are 35 days in a month.
(ii)Mathematics is difficult.
(iii)The sum of 5 and 7 is greater than 10.
(iv) The square of a number is an even number.
(v) The sides of a quadrilateral have equal length.
(vi) Answer this question.
(vii)The product of (-1) and 8 is 8.
(viii) The sum of all interior angles of a triangle is 180°.
(ix) Today is a windy day.
(x) All real numbers are complex numbers.

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

For each of the following compound statement first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
(ii) Square of an integer is positive or negative.
(iii) The sand heats up quickly in the sun and does not cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x² - x - 10 = 0.

Write the contrapositive and converse of the following statements.
(i) if x is a prime number, then x is odd.
(ii) if the two lines are parallel, then they do not intersect in the same plane.
(iii) Something is cold implies that it has low temperature. 
(iv) You cannot comprehend geometry if you do not know how to reason deductively.
(v) x is an even number implies that x is divisible by 4.

Show that the statement
p:"If x is a real number such that :x³+ 4x = 0 then x is 0" is true by
(i) direct method
(ii) method of contradiction
(iii) method of contrapositive.

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.

Prove that the statement
"If x\(\in \)R such that x³+ 7x = 0 then x = 0" is true by
(i) direct method
(ii) method of contradiction
(iii) method of contrapositive.

State the converse and contrapositive of each of the following statements:
(i) p: A positive integer is prime only if it has
no divisors other then 1 and itself.
(ii) q: I goto a beach whenever it is a sunny day.
(iii) r: if it is not outside, then you feel thirsty.

Check the validity of the statements given below by the method given against it
(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method)
(ii) q: If n is a real number with n > 3, then n² > 9 (by contradiction method)