To understand the correctness of the statements, let's break down the key terms mentioned in each statement.

Statement I: Failure of segregation of chromatids during cell cycle resulting in the gain or loss of whole set of chromosome in an organism is known as aneuploidy.

Aneuploidy refers to a condition in which the number of chromosomes is not an exact multiple of the haploid set. This typically results in the gain or loss of one or more chromosomes, not entire sets. Therefore, the definition provided in the statement is incorrect as it suggests a whole set of chromosomes, which is more aligned with polyploidy.

Statement II: Failure of cytokinesis after anaphase stage of cell division results in the gain or loss of a chromosome is called polyploidy.

Polyploidy is a condition in which an organism has more than two complete sets of chromosomes. This typically occurs due to the failure of cytokinesis, but it does not result in the gain or loss of a single chromosome, rather, it results in the addition of entire sets of chromosomes.

Given the explanations above, the correct answer would be:

Option D: Both Statement I and Statement II are false