Example of relations in Euclidean plane

Consider the set \(C = \{(x, y) : x^2 + y^2 = 1\}\). Then \(C\) is a relation of \(\mathbb{R} \times \mathbb{R}\).

Consider the set \(L = \{(x, y) : y = 2x - 1\}\). Then \(L\) is yet another relation of \(\mathbb{R} \times \mathbb{R}\).

Can you represent these two relations on the Euclidean 2D plane?

For \(C\), we can see that \(0^2 + 1^2 = 1\) and \(0^2 + (-1)^2 = 1\) so that \((0, 1), (0, -1) \in C\). Thus, the element \(0\) is linked with two distinct elements, namely, \(1\) and \(-1\). Note also that \(2\) is not linked to any element via the relation \(C\); i.e., we say that \(2\) is not \(C\)-related to any real numbers.

For \(L\), every real number \(x\) is \(L\)-related to only one real number \(y = 2x - 1\).