Question 2
In each of the following questions, there are two statements. State if the second is the negation of the first or not.
(i) Mumtaz is hungry. Mumtaz is not hungry.
(ii) Some cats are black. Some cats are brown.
(iii) All elephants are huge. One elephant is not huge.
(iv) All fire engines are red. All fire engines are not red.
(v) No man is a cow. Some men are cows.
We will check if the second statement is true exactly when the first statement is false.
Step 1 — Define Negation
Let's first understand what negation means. A statement's negation is its exact opposite. If the first statement is true, its negation must be false. If the first statement is false, its negation must be true. They cannot both be true. They cannot both be false.
Step 2 — Analyze Each Pair
(i) Mumtaz is hungry. / Mumtaz is not hungry.
If Mumtaz is hungry, the first statement is true. Then the second statement is false. If Mumtaz is not hungry, the first statement is false. Then the second statement is true. This pair is a negation.
(ii) Some cats are black. / Some cats are brown.
Suppose some cats are black, and some cats are brown. In this case, both statements are true. Since both can be true at the same time, the second is not the negation of the first. The negation of "Some cats are black" would be "No cats are black".
(iii) All elephants are huge. / One elephant is not huge.
"All elephants are huge" means every elephant is huge. Its negation means it is not true that all are huge. This means at least one elephant is not huge. "One elephant is not huge" means at least one is not huge. This pair is a negation.
(iv) All fire engines are red. / All fire engines are not red.
Let's imagine some fire engines are red, and some are yellow. Then "All fire engines are red" is false. Also, "All fire engines are not red" is false. Since both can be false at the same time, the second is not the negation of the first. The negation of "All fire engines are red" is "Some fire engines are not red".
(v) No man is a cow. / Some men are cows.
"No man is a cow" means it is false that any man is a cow. This is the same as "All men are not cows". The negation of "All are not P" is "Some are P". This pair is a negation.
Answer
(i) Yes, it is the negation. (ii) No, it is not the negation. (iii) Yes, it is the negation. (iv) No, it is not the negation. (v) Yes, it is the negation.
More questions in A1.4
State the negations for the following statements :
(i) Man is mortal.
(ii) Line is parallel to line .
(iii) This chapter has many exercises.
(iv) All integers are rational numbers.
(v) Some prime numbers are odd.
(vi) No student is lazy.
(vii) Some cats are not black.
(viii) There is no real number , such that .
(ix) 2 divides the positive integer .
(x) Integers and are coprime.
In each of the following questions, there are two statements. State if the second is the negation of the first or not.
(i) Mumtaz is hungry. Mumtaz is not hungry.
(ii) Some cats are black. Some cats are brown.
(iii) All elephants are huge. One elephant is not huge.
(iv) All fire engines are red. All fire engines are not red.
(v) No man is a cow. Some men are cows.