Question

Consider the subset \(R=(\mathbb{R} \times \mathbb{R})-\\{(x, x): x \in \mathbb{R}\\} \subseteq \mathbb{R} \times \mathbb{R} .\) What familiar relation on \(\mathbb{R}\) is this? Explain.

Short answer

The subset \((\mathbb{R} \times \mathbb{R})-\{(x, x): x \in \mathbb{R}\}\) is equivalent to the 'not equal to' relation on the set of all real numbers, \(\mathbb{R}\).

Step-by-step solution

Understanding the given subset \((\mathbb{R} \times \mathbb{R})-\{(x, x): x \in \mathbb{R}\}\)

This notation represents a subset of the Cartesian product \(\mathbb{R} \times \mathbb{R}\). Hence, this subset consists of all ordered pairs of real numbers, but excludes the pairs in which both components are the same real number, i.e., pairs of the form (x, x), where x is any real number.

Relating the subset to a familiar relation

Upon understanding the given subset, we can see it represents all the ordered pairs (x, y) where x and y are distinct real numbers. In terms of relations, this subset corresponds to the nonzero difference relation on the set of real numbers, \(\mathbb{R}\). In this relation, an ordered pair (x, y) is in the relation if and only if x - y ≠ 0 or equivalently x ≠ y.

Final explanation

In summary, the given subset \((\mathbb{R} \times \mathbb{R})-\{(x, x): x \in \mathbb{R}\}\) is equivalent to the 'not equal to' relation on the set of all real numbers. This is a familiar relation we often deal with in basic algebra and calculus.