In each of the following questions, there are two statements. State if the second is the negation of the first or not.
Mumtaz is hungry. Mumtaz is not hungry.
Yes, the second statement is the negation of the first. 'Mumtaz is not hungry' is obtained by inserting 'not' in the original statement, which gives the correct negation for simple statements.
Some cats are black. Some cats are brown.
No, the second statement is NOT the negation of the first. 'Some cats are brown' is a different statement altogether. The negation of 'Some cats are black' would be 'No cats are black' or 'It is false that some cats are black.'
All elephants are huge. One elephant is not huge.
Yes, the second statement is the negation of the first. The negation of 'All elephants are huge' means 'It is false that all elephants are huge,' which correctly translates to 'There is at least one elephant that is not huge' or 'One elephant is not huge.'
All fire engines are red. All fire engines are not red.
No, the second statement is NOT the negation of the first. Simply inserting 'not' in statements beginning with 'All' causes confusion. 'All fire engines are not red' could imply all fire engines are of some other colour. The correct negation would be 'Some fire engines are not red' or 'Not all fire engines are red.'
No man is a cow. Some men are cows.
Yes, the second statement is the negation of the first. The negation of 'No man is a cow' is 'It is false that no man is a cow,' which correctly means 'There is at least one man who is a cow' or 'Some men are cows.'
Explanation
The textbook provides a Working Rule for negation: (i) First write the statement with a 'not', and (ii) if there is confusion, make suitable modifications, especially for statements with 'All' or 'Some'. The key criterion is that the negation of a statement p (denoted ~p) must be false when p is true, and true when p is false. For statements beginning with 'All', simply inserting 'not' creates ambiguity. The correct negation of 'All A are B' is 'Some A are not B' or 'Not all A are B', meaning 'There exists at least one A that is not B.'