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

(i) x + y = y + x is true for every real number x and y.

(ii) There exists a real number x and y for which x + y = y + x.

For each of the following compound statements 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.

Identify the quantifier in the following statements and write the negation of the statements.

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

(ii) For every real number x, x is less then x + 1.

(iii) There exists a capital for every state in India.

State whether the "OR" used in the following statements is "exclusive" or "inclusive". Give reasons for your answer.

(i) Sun rises or moon sets.

(ii) To apply for a driving license, you should have a ration card or a passport.

(iii) A lady gives birth to a baby boy or baby girl.

(iv) All integers are positive or negative.

Rewrite the following statement with "if-then" in five different ways conveying the same meaning. If a natural number is odd, then its square is also odd.