what is the property that distinguishes finite sets from infinite sets (give examples of each to accompany explaination).
finite sets are countable. Infinite sets are not.
so what would be an example of an infinite set? one that never ends?
yes. 1,2,3,4,... is an infinite set.
1,2,3,4,5,6 is a finite set.
Countable can also refer to infinite sets that can be put into one to one correspondence with the integers.
E.g. the set of all fractions is countable, while the set of reals is not countable.
You are right
To determine whether a set is finite or infinite, you need to understand the concept of countability. A finite set is one that has a fixed number of elements and can be counted or listed explicitly. An infinite set, on the other hand, has an uncountable number of elements and cannot be listed or counted exhaustively.
For example, let's consider the set {1, 2, 3, 4, 5, 6}. This set has six elements, and we can explicitly list all of them. Therefore, it is a finite set.
In contrast, the set {1, 2, 3, 4, ...} is an infinite set. This set includes all natural numbers starting from 1 and continuing indefinitely. No matter how far you count, there will always be another number in the sequence. Hence, this set is infinite.
Another example of an infinite set is the set of all fractions (or rational numbers). Although the fractions can be represented by numbers, there are infinitely many fractions between any two integers. Despite being countable in a sense that each fraction can be assigned a unique integer, the set is infinite because it goes on indefinitely.
On the other hand, the set of real numbers is an example of an uncountable infinite set. Unlike fractions, real numbers cannot be assigned a unique integer and thus cannot be counted one by one. The real numbers include not only whole numbers and fractions but also irrational numbers such as √2 or π. The infinite nature of the real numbers becomes apparent because we can always find more real numbers between any two given real numbers.
In summary, the property that distinguishes finite sets from infinite sets is countability. Finite sets have a fixed number of elements and are countable, while infinite sets have an uncountable number of elements and cannot be listed or counted exhaustively.