Basics

The given condition satisfies all the natural numbers, but the thing mentioned in the question is for all non-negative integers, where non-negative integers are from $0 \text{ to } \infty$ . All the numbers satisfy except zero.

$\large \displaystyle \because 0^0 \text{is not defined} \ngeq 0$ .

For natural numbers, $\large \displaystyle x^x \geq 0$ .

$0^0$ is undetermined, so the equation does not hold for all non-negative integers. Hence the answer is false.