Appendix 1: Proofs in Mathematics | A1.4

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.

Solution

Understand the Question

The negation of a statement is a proposition that has the exact opposite truth value in all cases.
A statement and its negation cannot both be true at the same time, nor can they both be false at the same time.
To check whether the second statement is the negation of the first:
- The negation of "All $A$ are $B$ " is "Some $A$ are not $B$ " (at least one $A$ is not $B$ ).
- The negation of "No $A$ is $B$ " is "Some $A$ are $B$ ".

(i) Mumtaz is hungry. Mumtaz is not hungry.

Step 1 · Check the Truth Values

If "Mumtaz is hungry" is true, then "Mumtaz is not hungry" is false. If "Mumtaz is hungry" is false, then "Mumtaz is not hungry" is true.

Since one is true precisely when the other is false, the second statement is the negation of the first.

Answer

(i) Yes, it is the negation.

(ii) Some cats are black. Some cats are brown.

Step 1 · Check the Truth Values

Both statements can be simultaneously true (e.g., if there exist both black cats and brown cats).

The true negation of "Some cats are black" is "No cats are black".

Answer

(ii) No, it is not the negation.

(iii) All elephants are huge. One elephant is not huge.

Step 1 · Check the Truth Values

"All elephants are huge" is false if and only if there exists at least one elephant that is not huge.

Therefore, "One elephant is not huge" correctly negates "All elephants are huge".

Answer

(iii) Yes, it is the negation.

(iv) All fire engines are red. All fire engines are not red.

Step 1 · Check the Truth Values

If some fire engines are red and some are yellow, both statements are false at the same time.

The negation of "All fire engines are red" is "Some fire engines are not red".

Answer

(iv) No, it is not the negation.

(v) No man is a cow. Some men are cows.

Step 1 · Check the Truth Values

"No man is a cow" means all men are not cows. This statement is false if and only if there is at least one man who is a cow.

Therefore, "Some men are cows" is the exact negation.

Answer

(v) Yes, it is the negation.

Common Mistakes

Opposite vs Negation: Negating "All are $P$ " is not "All are not $P$ ", but rather "At least one is not $P$ " ("Some are not $P$ ").
Simultaneous Truth: Thinking statements about different properties (like black and brown cats) are negations simply because they describe different categories, ignoring that both can be true simultaneously.

More questions in A1.4

State the negations for the following statements :

(i) Man is mortal.

(ii) Line $l$ is parallel to line $m$ .

(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 $x$ , such that $\sqrt{x} = -1$ .

(ix) 2 divides the positive integer $a$ .

(x) Integers $a$ and $b$ 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.