Can someone explain to me why the intersection of two infinite sets is not always an infinite set, but the union of two infinite sets is?
Take the infinite set (0, odd numbers). Then consider the infinite set (0. even numbers).
The intersection is not infinite, it contains just 0
But the union is infinite, as it contains 0, all odd, and all even
That makes sense. Thank you!
Certainly! Let's start by understanding the concepts of intersection and union of sets.
The intersection of two sets, denoted as A ∩ B, consists of the elements that are common to both sets A and B. In other words, it includes all the elements that are present in both sets.
The union of two sets, denoted as A ∪ B, consists of all the elements that are present in either set A or set B, or both. In simple terms, it combines all the elements from both sets without duplicating them.
Now, to understand why the intersection of two infinite sets is not always infinite, we need to consider the properties of infinite sets.
An infinite set is defined as a set with an unlimited number of elements. It continues indefinitely without end. For example, the set of all positive integers {1, 2, 3, 4, ...} is an infinite set.
When we take the intersection of two infinite sets, it means we are looking for the common elements between them. While it is possible that their intersection could also be infinite, it is not always the case.
To see an example, let's consider two infinite sets:
Set A: {2, 4, 6, 8, ...} (set of even numbers)
Set B: {3, 6, 9, 12, ...} (set of multiples of 3)
The intersection of A and B is the set of numbers that are both even and multiples of 3. In this case, the only number that satisfies this condition is 6. Therefore, their intersection is {6}, which consists of a single element and is not infinite.
In contrast, when we take the union of two infinite sets, we are combining all the elements present in both sets. Since both sets individually have an unlimited number of elements, their union will also have an unlimited number of elements, making it infinite.
For example, the union of sets A and B from our previous example would include all even numbers and all multiples of 3: {2, 3, 4, 6, 8, 9, 10, 12, ...} – and this set continues indefinitely, making it an infinite set.
In summary, the intersection of two infinite sets might not always be an infinite set because it depends on the common elements between them. On the other hand, the union of two infinite sets is always infinite because it combines all the elements from both sets.