Let \(\mathrm{X}=\\{(\mathrm{x}, \mathrm{y}, \mathrm{z}) / \mathrm{x}, \mathrm{y}, \mathrm{z} \in \mathrm{N}, \mathrm{x}+\mathrm{y}+\mathrm{z}=10, \mathrm{x}<\mathrm{y}<\mathrm{z}\\}\) and \(\mathrm{Y}=\\{(\mathrm{x}, \mathrm{y}, \mathrm{z}) / \mathrm{x}, \mathrm{y}, \mathrm{z} \in \mathrm{N}, \mathrm{y}=|\mathrm{x}-\mathrm{z}|\\}\) then \(\mathrm{X} \cap \mathrm{Y}\) is equal to (a) \(\\{(2,3,5)\\}\) (b) \(\\{(1,4,5)\\}\) (c) \(\\{(5,1,4)\\}\) (d) \(\\{(2,3,5),(1,4,5)\\}\)

Natural numbers, often represented by the symbol \( N \), form the building block of mathematics that we encounter in everyday life. This set includes all the positive integers starting from 1, 2, 3, and so on, infinitely. When we're dealing with set theory problems that involve natural numbers, it's crucial to remember that 0 is not included in this set, as it represents the absence of quantity; therefore, the smallest natural number is 1.

The significance of natural numbers in set theory problems, such as the exercise at hand, cannot be understated. Here, each component within the ordered triples (x, y, z) needs to be a natural number. This means each element must be positive and without any fractional or decimal component. This particular constraint shapes the possible solutions for the set and adheres to the basic attribute of natural numbers.

The concept of set intersection is pivotal in understanding how different sets interact with each other. Symbolically, the intersection is denoted by \( \cap \), and it essentially allows us to find the common elements between two sets. If we have two sets, A and B, the intersection of A and B, written as \( A \cap B \), will include all the elements that A and B share. If there are no common elements, we say that the intersection is an empty set, represented by \( \varnothing \).

In the provided exercise, to determine \( X \cap Y \), we must identify ordered triples that satisfy the conditions of both Set X and Set Y. It's like finding a middle ground where both sets' requirements meet. As seen in the solution, once the components satisfying Set X and Set Y individually are found, these components are cross-checked to single out the overlapping ones—the resultant intersection being the shared elements evidences the principle of set intersection.

Incorporating the concept of absolute value in set problems adds another layer to the condition that set elements must satisfy. The absolute value function, usually denoted as \( |a| \), measures the distance of a number 'a' from 0 on the number line without considering the direction. This means the absolute value is always non-negative since distance cannot be negative.

When the condition \( y = |x - z| \) is applied in Set Y within our exercise, we're effectively stating that the value of y should be equal to the non-negative difference between x and z. This plays a crucial role in constraining the set to only those ordered triples where the second element reflects the absolute magnitude of the difference between the first and third elements. As the solution indicates, this absolute value condition eliminates many potential elements from Set Y that do not conform to this specific rule.