For \(\mathrm{n}, \mathrm{m} \in \mathrm{N} \mathrm{n} / \mathrm{m}\) means that \(\mathrm{n}\) is a factor of \(\mathrm{m}\), the relation/is (a) reflexive and symmetric (b) transitive and symmetric (c) reflexive transitive and symmetric (d) reflexive transitive and not symmetric

Reflexivity is a property of mathematical relations that tells us something quite intuitive: every element is related to itself. When dealing with numbers, this concept is often visualized in the context of division. For instance, when we speak of the relation \( n / n \) where \( n \) is a natural number, reflexivity is showcased by the fact that any number divided by itself is equal to 1 (assuming we exclude division by zero).

The act of checking reflexivity in a relation is quite straightforward. You only need to confirm that for every element you're considering—in this case, for every number \( n \)—the relation \( n \) relates to itself. If it does, as it does in our division example where every number can divide itself, then the relation is reflexive. This concept is foundational in mathematics, as it is one of the properties that defines equivalence relations, which are key in partitioning sets into equivalence classes.

Symmetry in relations refers to a scenario where if an element \( a \) is related to an element \( b \) then \( b \) is also related to \( a \). This mutual relationship doesn't hold true in all cases, especially not in the division of numbers.

Symmetry is easily understood with an example: If \( n \) is a factor of \( m \) (denoted as \( n / m \) in the exercise), symmetry would require that \( m \) also be a factor of \( n \) (\( m / n \) ). However, when dealing with division, the numbers don’t always work out so equivalently. A larger number cannot be a factor of a smaller number, so in the context of division among natural numbers, the relation is inherently not symmetric. This lack of symmetry is crucial in the distinction between different types of relations in mathematics.

Transitivity, another pillar in the understanding of mathematical relations, takes us a step beyond individual elements to consider how pairs of related items interact with a third. A relation is transitive if whenever \( a \) relates to \( b \) and \( b \) relates to \( c \), then \( a \) also relates to \( c \).

In our division example, when we say \( n \), \( m \) and \( k \) are natural numbers such that \( n \) is a factor of \( m \) and \( m \) is a factor of \( k \), transitivity ensures that \( n \) is also a factor of \( k \). This is demonstrated by multiplying the unique factors of \( k \) and \( m \) together to show that you can then multiply \( n \) by this product to get \( k \). Transitivity is a key feature in defining order relations and many algebraic structures.

The final concept lying behind our exercise is number theory, which is a branch of pure mathematics devoted primarily to the study of the integers and their properties. Within number theory, concepts such as divisibility play a significant role. Divisibility itself is the foundation of understanding factors and multiples—core concepts of our exercise about division as a relation.

Number theory tackles questions about prime numbers, congruences, and diophantine equations, to name a few areas. It finds usefulness in cryptography and computer science, among other fields. The acknowledgement that a number can have factors and that these factors interact with each other through relations like divisibility emphasizes the relevance of number theory in even basic exercises like the one we’ve been discussing.