13.3.2. Non-negative Integers under Addition

Let $\mathbb{Z}_{\geq 0}$ be the set of non-negative integers (i.e., 0, 1, 2, 3, ...), and let the operation be addition (+). The closure property states that the sum of any two non-negative integers is also a non-negative integer. The identity element for addition is 0, as adding 0 to any non-negative integer $x$ results in $x$.

The closure property tells us that if we add any two non-negative integers, the result is still a non-negative integer. For example, adding 2 and 3 gives us 5, which is also a non-negative integer. The identity property informs us that there is a special number that, when added to any integer in our set, does not change it. In this case, that number is 0. For instance, if we have 4 and we add 0 to it, we still have 4.

Imagine you have a jar containing candies (where each candy represents a non-negative integer). If you take out 3 candies and later you put in 2 more, you will always have some candies left. If you don’t add any returns (0 candies), the number of candies remains unchanged.

The associativity property states that when adding any three non-negative integers, it does not matter how they are grouped. For any integers $a, b, c \in \mathbb{Z}_{\geq 0}$, it holds that $(a + b) + c = a + (b + c)$.

The associativity property means that when adding multiple numbers, the way we group them (which pairs we add together first) does not affect the final sum. For example, if we add 1, 2, and 3, we could first sum 1 and 2 to get 3, then add 3 to the result, or we could sum 2 and 3 first to get 5, then add 1. In both cases, the final sum is the same: 6.

Consider three friends at a candy store. If they decide to buy candies in two different ways—First, two of them buy some candies together and then they all meet the third one, or each buys separately and then combines their candies later—the total number of candies remains the same regardless of the order they decide.

Despite the fact that the closure and identity properties hold for non-negative integers under addition, the requirement for an inverse element fails. The inverse of a non-negative integer $x$ is $-x$, which is not in the set of non-negative integers if $x$ is greater than 0.

An inverse element, in context of addition, is a number that, when added to a given number, yields the identity element (0). For example, the inverse of 2 is -2, because 2 + (-2) = 0. However, -2 is not part of the set of non-negative integers. Hence, for any positive integer in our set, we cannot find a corresponding inverse that fits within the same set.

Think of having 10 apples; if you want to take apples away to bring your count back to zero, you would need to take away more than you have, which isn't possible in our non-negative realm. Therefore, one cannot neutralize their count using the numbers within their own group.

Since the requirement for an inverse element is not satisfied, the set of non-negative integers under addition does not form a group, despite the first three group axioms being met.

In summary, while non-negative integers satisfy closure, identity, and associativity properties, the lack of inverses prevents them from being classified as a group. This is a crucial point to remember: all four properties must hold true for a set and operation combination to be considered a group. If even one axiom fails, they do not form a group.

Think of a group of friends trying to play a game where everyone needs a partner. If one friend can't find anyone to play with (no inverse), they can't fully participate in the game, highlighting the concept of completeness required to form a group.

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