eibon said:well its the first time I am taking an math class like this and I am just not sure if I am doing things right
Denumerability refers to the property of a set to be able to be counted or enumerated. In other words, a set is denumerable if its elements can be arranged in a sequence and each element can be assigned a unique natural number.
Proving the denumerability of a set is important in mathematics as it helps establish the cardinality (size) of the set. In this case, proving the denumerability of positive multiples of three greater than 100 helps us understand the infinite nature of this set and its relationship to other infinite sets.
To prove denumerability, we need to show that every element in the set can be paired with a unique natural number. In this case, we can do so by showing that for every natural number n, there exists a positive multiple of three greater than 100 that can be expressed as 3n + 100.
There are different methods for proving denumerability, depending on the set in question. In general, the most common methods involve constructing a bijection (one-to-one correspondence) between the set and the set of natural numbers, or using the diagonalization argument.
No, a set cannot be both denumerable and uncountable. A set is either denumerable (countable) or uncountable, as these are two mutually exclusive properties. A set is uncountable if it cannot be put into a one-to-one correspondence with the set of natural numbers.