Consider the set&nbsp;\(C = \{(x, y) : x^2 + y^2 = 1\}\). Then&nbsp;\(C\) is a relation of \(\mathbb{R} \times \mathbb{R}\).
Consider the set&nbsp;\(L = \{(x, y) : y = 2x - 1\}\). Then&nbsp;\(L\) is yet another relation of \(\mathbb{R} \times \mathbb{R}\).
Can you represent these two relations on the Euclidean 2D plane?
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1770372741148-jqraaayunl.png" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Figure 1: Representations of \(C\) and \(L\)." class="aligncenter custom-width" /></figure></div>
For&nbsp;\(C\), we can see that&nbsp;\(0^2 + 1^2 = 1\) and \(0^2 + (-1)^2 = 1\) so that&nbsp;\((0, 1), (0, -1) \in C\). Thus, the element \(0\) is linked with two distinct elements, namely, \(1\) and \(-1\). Note also that&nbsp;\(2\) is not linked to any element via the relation \(C\); i.e., we say that&nbsp;\(2\) is not \(C\)-related to any real numbers.
For&nbsp;\(L\), every real number \(x\) is \(L\)-related to only one real number \(y = 2x - 1\).

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 \(y = 2x - 1\).

Functions

International Baccalaureate Higher Level Mathematics

Example of relations in Euclidean plane

Example of relations in Euclidean plane