Appendix 1: Proofs in Mathematics | A1.4

Question 1

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.

Solution

Understand the Question

The negation of a statement is formed by denying what the statement asserts; it is true whenever the original statement is false, and false whenever the original statement is true.
Standard rules for negating statements:
- Simple statements: Introduce or remove "not" / "does not".
- "All $A$ are $B$ ": Negation is "Some $A$ are not $B$ " (or "It is not the case that all $A$ are $B$ ").
- "Some $A$ are $B$ ": Negation is "No $A$ is $B$ ".
- "No $A$ is $B$ ": Negation is "Some $A$ are $B$ ".
- "Some $A$ are not $B$ ": Negation is "All $A$ are $B$ ".
- "There is no...": Negation is "There exists / There is...".

(i) Man is mortal.

Answer

(i) Man is not mortal.

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

Answer

(ii) Line $l$ is not parallel to line $m$ .

(iii) This chapter has many exercises.

Answer

(iii) This chapter does not have many exercises.

(iv) All integers are rational numbers.

Answer

(iv) Some integers are not rational numbers.

(v) Some prime numbers are odd.

Answer

(v) No prime numbers are odd.

(vi) No student is lazy.

Answer

(vi) Some students are lazy.

(vii) Some cats are not black.

Answer

(vii) All cats are black.

(viii) There is no real number $x$ , such that $\sqrt{x} = -1$ .

Answer

(viii) There is a real number $x$ , such that $\sqrt{x} = -1$ .

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

Answer

(ix) 2 does not divide the positive integer $a$ .

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

Answer

(x) Integers $a$ and $b$ are not coprime.

Common Mistakes

Negating "All" with "No": The negation of "All $A$ are $B$ " is not "No $A$ is $B$ ", but "Some $A$ are not $B$ " (it only takes one counterexample to disprove an "all" claim).
Negating "No" with "All": The negation of "No $A$ is $B$ " is "Some $A$ are $B$ ", not "All $A$ are $B$ ".

Question 1

More questions in A1.4