Prove or disprove in the below questions:
$1.$ Let $a$ and $b$ be real numbers. Then $(a+b)^3=a^3+b^3$ implies $a=0$ OR $b=0$.
Disproof. Let $a=-1$ and $b=1$. Then, we also get $(a+b)^3=a^3+b^3=0$.
$2.$ Let $a$ and $b$ be real numbers. Then $(a+b)^3=a^3+b^3$ implies $a=0$ AND $b=0$.
Disproof. Let $a=-1$ and $b=1$. Then, we also get $(a+b)^3=a^3+b^3=0$.
Can you say what is the difference between question-1 and question-2?
And can you check my proofs?
On
Your two disproofs are correct. To clarfy difference, for example if you could have said like this (and suppose it is true for $x\neq 0$, I know it is not true but just suppose)
Disproof: Let $a=x$, $b=0$ then we get $(a+b)^3=a^3+b^3=0$
Then the disproof would have worked for question 2, but not 1
On
The selected counterexample disproves both claims. Yet not all counterexamples will disprove both claims.
To demonstrate the difference between the claims consider this:
Disproof: Let $a=0$ or $b=0$, but not both, then we have $(a+b)^3=a^3+b^3$
This counterexample disproves the claim that the equality implies $a=0$ and $b=0$.
However, it clearly does not disprove the claim that the equality implies $a=0$ or $b=0$.
On
The complete solution of the equation is: $$(a+b)^3=a^3+b^3 \iff \{(a,b)\in R^2|a=0 \ \text{or} \ b=0 \ \text{or} \ a=-b\}.$$ And note: $$\begin{align}A&=\{(a,b)\in R^2|a=0 \ \text{and} \ b=0\};\\ B&=\{(a,b)\in R^2|a=0 \ \text{or} \ b=0\}; \\ C&=\{(a,b)\in R^2|a=0 \ \text{or} \ b=0 \ \text{or} \ a=-b\};\\ A& \subset B \subset C.\end{align}$$ You can disprove the elements of the smaller set by taking the elements of the bigger set. For example: $$(a,b)=(0,5)\notin A; (a,b)=(0,5)\in B \Rightarrow (0+5)^3=0^3+5^3 \ \ \checkmark \\ (a,b)=(1,-1)\notin B; (a,b)=(1,-1)\in C \Rightarrow (1-1)^3=1^3+(-1)^3 \ \ \checkmark \\$$
Well, in that specific example the answers are the same. Still, these are two different questions. For example $(a+b)^2=a^2+b^2$ implies $a=0$ or $b=0$ but it doesn't imply that both must be zeros.
And yes, your solution is correct.