Some even integers are prime.

Form the biconditional statement p ↔ q, where

p : A triangle is an equilateral triangle.

q : All three sides of a triangle are equal.

Write down the contrapositive of the following statements:

(i) If x = y and y = 3, then x = 3.

(ii) If n is a natural number, then n is an integer.

(iii) If all three sides of a triangle are equal, then the triangle is equilateral.

(iv) If x and y are negative integers, then xy is positive.

(v) If natural number n is divisible by 6, then n is divisible by 2 and 3.

(vi) If it snows, then the weather will be cold.

(vii) If x is a real number such that 0 < x < 1, then x² < 1.

Rewrite each of the following statements in the form of conditional statements

(i) The square of an odd number is odd.

(ii) You will get a sweet dish after the dinner.

(iii) You will fail, if you will not study.

(iv) The unit digit of an integer is 0 or 5 if it is divisible by 5.

(v) The square of a prime number is not prime.

(vi) 2b = a + c, if a, b and c are in A.P.

Write down the converse of following statements :

(i) If a rectangle ‘R’ is a square, then R is a rhombus.

(ii) If today is Monday, then tomorrow is Tuesday.

(iii) If you go to Agra, then you must visit Taj Mahal.

(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.

(v) If all three angles of a triangle are equal, then the triangle is equilateral.

(vi) If x : y = 3 : 2, then 2x = 3y.

(vii) If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.

(viii) If x is zero, then x is neither positive nor negative.

(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

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.

Write the negation of the following statements:

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

Write the negation of each of the following statements:

(i) For every x ∈ N, x + 3 > 10

(ii) There exists x ∈ N, x + 3 = 10

Negate each of the following statements:

(i) All the students complete their homework.

(ii) There exists a number which is equal to its square.

Check whether the following pair of statements is a negation of each other. Give reasons for your answer.

(i) a + b = b + a is true for every real number a and b.

(ii) There exist real numbers a and b for which a + b = b + a.