Equivalence Relation on Ordered Pairs

Let $R$ be the relation on the set of ordered pairs of positive integers such that $(a, b) \, R \, (c, d)$ if and only if $ad = bc$. Show that $R$ is an equivalence relation.

This problem involves demonstrating that a relation on a set is an equivalence relation. An equivalence relation is one that is reflexive, symmetric, and transitive. These properties are fundamental in discrete mathematics as they help classify and categorize elements within sets into equivalence classes. The problem is set in the context of ordered pairs of positive integers with a relation defined by the equality of cross products, specifically: if and only if the product of the outer variables equals the product of the inner ones. Identifying and proving that a given relation is an equivalence relation is a crucial skill when tackling problems in set theory and functions, as well as abstract algebra and number theory.

The strategy for showing that a relation is an equivalence relation involves breaking down the problem into three parts: proving reflexivity, symmetry, and transitivity. Reflexivity requires showing that every element is related to itself. Symmetry involves demonstrating that if one element is related to another, then the reverse holds true as well. Lastly, transitivity is about proving that if one element is related to a second, which in turn is related to a third, then the first element must be related to the third. Each of these properties relates back to the fundamental idea of equivalence, where elements are considered equivalent if they share certain attributes or relationships, allowing for partitioning of a set into distinct, non-overlapping subsets where each party in a subset is equivalent to each other based on the defined relation. This process of verifying equivalence relation properties not only strengthens comprehension of fundamental mathematical concepts but also enhances abstract thinking and proof-writing skills crucial at this level.