Question

Calculate the cardinal sum \(\mathrm{N}_{\mathrm{O}}+\mathrm{N}_{\mathrm{O}}\) where \(\mathrm{N}_{\mathrm{O}}=\) card \(\mathrm{N}\) and \(\mathrm{N}\) is the set of natural numbers.

Short answer

The cardinal sum NO + NO, where NO is the cardinality of the set N (the set of natural numbers) is ∞.

Step-by-step solution

Understand the given information

We are given that the set N is the set of natural numbers, which are the numbers 1, 2, 3, ... (continuing to infinity). We are asked to find the cardinal sum NO + NO, where NO is the cardinality of the set N (meaning the number of elements in the set N).

Determine the cardinality of set N

N is the set of natural numbers which include all positive integers. The set of natural numbers, N, is an infinite set, as there is an infinite amount of positive integers. So, the cardinality (NO) of set N is infinite, which can be denoted as ∞.

Calculate the cardinal sum

We are asked to find the cardinal sum NO + NO. We found the cardinality (NO) of set N to be infinite (∞) in step 2. Therefore, the calculation for the cardinal sum looks like this: ∞ + ∞. Since infinity plus infinity is still infinity, the final answer for the cardinal sum is ∞.

Final Answer

The cardinal sum NO + NO, where NO is the cardinality of the set N (the set of natural numbers) is ∞.

Key concepts

Cardinality of a Set

The concept of 'cardinality of a set' refers to the number of elements contained within a set. This numerical value is significant in the realm of mathematics as it provides a measure of a set's size. When considering finite sets, those with a limited number of elements, the cardinality is simply the count of distinct elements.

For example, the set \( A = \{2, 4, 6, 8\} \) has a cardinality of 4 because there are four elements in the set. In computing cardinality, duplicates are not counted; it's the number of unique items that matters. In dealing with the cardinality of infinite sets, special rules apply because one cannot count their elements in a traditional sense. Understanding that infinite sets have an unbounded number of elements is crucial to grasping more advanced mathematical concepts.

Natural Numbers

The set of natural numbers \(\mathbb{N}\) includes all positive integers starting from 1 and continuing indefinitely. This sequence goes: 1, 2, 3, and so forth. Natural numbers are foundational in various branches of mathematics and are the building blocks of arithmetic and number theory.

Natural Numbers and Cardinality

When considering the cardinality of the set of natural numbers, it's pivotal to understand that although we cannot count to infinity, we denote the size of this set as 'infinite'. This infinite nature puts the set of natural numbers, along with other similarly unbounded sets, into a special category in terms of size and the operations you can perform with them.

Infinite Set

An 'infinite set' is one with no last element; it goes on forever. Unlike a finite set where you can list all the elements, an infinite set continues indefinitely without an endpoint. This might make it seem daunting, but mathematicians have developed ways to discuss these sets without needing to list every element.

In set theory, a branch of mathematics, infinite sets can be compared with each other to discuss their relative sizes in terms of cardinality. A familiar example of an infinite set is the set of natural numbers \(\mathbb{N}\), which includes all positive integers. This set is known to be 'countably infinite' because the elements can be associated with the set of positive integers. An important aspect of infinite sets is that adding elements or even adding an entire infinite set of the same size doesn't change their cardinality; it remains infinite.