Determine whether the argument used to check the validity of the following statement is correct:

p: “If x² is irrational, then x is rational.”

The statement is true because the number x² = π² is irrational, therefore x = π is irrational.

Determine whether in the given statements an inclusive “OR” or exclusive “OR” is used.

(i) Sun rises or moon sets

(ii) Two lines intersect at a point or are parallel.

(iii) The school is closed if it is holiday or a Sunday.

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 than x + 1.

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

Rewrite each of the following statements in the form “p if and only is q.”

(i) p : If you watch television, then your mind is free, and if your mind is free, then you watch television.

(ii) q : If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral is a rectangle, then it is equiangular.

(iii) r : For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly.

(iv) s : If a tumbler is half empty, then it is half full, and if a tumbler is half full, then it is half empty.

Write the component statements of the following compound statements and check whether the compound statement is true or false:

(i) To enter into a public library children need an identification card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

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

(iv) x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0.

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