State the negations for the following statements:
Man is mortal.
Man is not mortal.
Line is parallel to line .
Line is not parallel to line .
This chapter has many exercises.
This chapter does not have many exercises.
All integers are rational numbers.
It is false that all integers are rational numbers.
Some prime numbers are odd.
No prime numbers are odd.
No student is lazy.
Some students are lazy.
Some cats are not black.
All cats are black.
There is no real number , such that .
There is at least one real number , such that .
2 divides the positive integer .
2 does not divide the positive integer .
Integers and are coprime.
Integers and are not coprime.
Explanation
To find the negation of a statement, we use the working rule provided in the text: first write the statement with a 'not', and then make suitable modifications for statements involving 'All' or 'Some'. For simple statements like (i), (ii), (iii), (ix), and (x), we simply insert 'not' to deny the assertion. For statements involving quantifiers, we apply specific logic: the negation of 'All A are B' is 'Some A are not B' (as seen in the 'All teachers are female' example); the negation of 'Some A are B' is 'No A are B'; the negation of 'No A are B' is 'Some A are B' (as seen in the 'No man is a cow' example); and the negation of 'Some A are not B' is 'All A are B' (as seen in the 'Some horses are not brown' example). For existential statements like (viii), the negation of 'There is no...' is 'There is at least one...'.