Let and be real numbers such that . Then which of the following statements are true? Justify your answers. (i) Both and must be zero. (ii) Both and must be non-zero. (iii) Either or must be non-zero.

Understand the Question For real numbers and , the zero-product property states that if and only if at least one of the factors is zero ( or ). Conversely, their product if and only if both and are non-zero ( and ). Part (i) Both and must be zero. Step 1 · Check if Both Numbers Can Be Zero If and , their product is This contradicts the given condition . Answer > (i) False Part (ii) Both and must be non-zero. Step 1 · Check if Both Numbers Must Be Non-Zero If either or , the product becomes . Therefore, for , both and must be non-zero ( and ). Answer > (ii) True Part (iii) Either or must be non-zero. Step 1 · Check if at Least One Number Must Be Non-Zero If both and , their product is This contradicts . Since both numbers must be non-zero, it is guaranteed that at least one of them (either or ) is non-zero. Answer > (iii) True Common Mistakes Negating Zero-Product Property: Confusing the condition for (at least one is zero) with (both must be non-zero). Logic of "Either/Or": In mathematics, "either or is non-zero" means at least one is non-zero (inclusive OR). Since both and , this statement is logically true.

Appendix 1: Proofs in Mathematics | A1.1

Question 3

Let $a$ and $b$ be real numbers such that $ab \neq 0$ . Then which of the following statements are true? Justify your answers.

(i) Both $a$ and $b$ must be zero.

(ii) Both $a$ and $b$ must be non-zero.

(iii) Either $a$ or $b$ must be non-zero.

Solution

Understand the Question

For real numbers $a$ and $b$ , the zero-product property states that $ab = 0$ if and only if at least one of the factors is zero ( $a = 0$ or $b = 0$ ).
Conversely, their product $ab \neq 0$ if and only if both $a$ and $b$ are non-zero ( $a \neq 0$ and $b \neq 0$ ).

(i) Both $a$ and $b$ must be zero.

Step 1 · Check if Both Numbers Can Be Zero

If $a = 0$ and $b = 0$ , their product is

\begin{aligned} ab &= 0 \times 0 \\ &= 0 \end{aligned}

This contradicts the given condition $ab \neq 0$ .

Answer

(i) False

(ii) Both $a$ and $b$ must be non-zero.

Step 1 · Check if Both Numbers Must Be Non-Zero

If either $a = 0$ or $b = 0$ , the product becomes $ab = 0$ .

Therefore, for $ab \neq 0$ , both $a$ and $b$ must be non-zero ( $a \neq 0$ and $b \neq 0$ ).

Answer

(ii) True

(iii) Either $a$ or $b$ must be non-zero.

Step 1 · Check if at Least One Number Must Be Non-Zero

If both $a = 0$ and $b = 0$ , their product is

\begin{aligned} ab &= 0 \times 0 \\ &= 0 \end{aligned}

This contradicts $ab \neq 0$ . Since both numbers must be non-zero, it is guaranteed that at least one of them (either $a$ or $b$ ) is non-zero.

Answer

(iii) True

Common Mistakes

Negating Zero-Product Property: Confusing the condition for $ab = 0$ (at least one is zero) with $ab \neq 0$ (both must be non-zero).
Logic of "Either/Or": In mathematics, "either $a$ or $b$ is non-zero" means at least one is non-zero (inclusive OR). Since both $a \neq 0$ and $b \neq 0$ , this statement is logically true.

Question 3

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