(For comparison, when sampling with replacement, observation of a particular value has no effect on how likely it is to observe that value again.)

From this perspective, the case labeled "Surjective f" is somewhat strange: Essentially, we keep sampling with replacement until we've chosen each item at least once.

In the former case (sampling with replacement), once we've chosen an item, we put it back in the population, so that we might choose it again.

Partition into subsets with each element of belonging to one of the with equal probability (sampling with replacement)

Imagine a population of X items (or people), of which we choose N. Two different schemes are normally described, known as "sampling with replacement" and "sampling without replacement".

Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the odds of choosing the same individual twice is low.

In terms of probability distributions, sampling with replacement where ordering matters is comparable to describing the joint distribution of N separate random variables, each with an X-fold categorical distribution.

Bootstrapping is a statistical method for estimating the sampling distribution of an estimator by sampling with replacement from the original sample, most often with the purpose of deriving robust estimates of standard errors and confidence intervals of a population parameter like a mean, median, proportion, odds ratio, correlation coefficient or regression coefficient.