Show that the following statement is true

“The integer n is even if and only if n² is even”.

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.

Given below are two statements

p : 25 is a multiple of 5.

q : 25 is a multiple of 8.

Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

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).