Reflections in Relations?

📌 CONCEPT: A reflection in a relation refers to a transformation where the elements of the relation are reflected over a certain axis or line, often resulting in a symmetric or mirrored image.

📐 RULE / FORMULA: To show that $(P ext{ ∪ } Q) R (P ext{ ∩ } Q),$ we need to prove that $f R g$ for all $f ext{ ∪ } g$ and $f ext{ ∩ } g.$

💡 WORKED EXAMPLE: Let's consider $P(x) = 2x^2 + 1$ and $Q(x) = rac{2x^2 + 1}{x}.$ Clearly, $P(x) eq Q(x)$ for all real $x.$ However, for $x eq 0,$ we have $P(x) ext{ ∩ } Q(x) = (2x^2 + 1) ext{ ∩ } rac{2x^2 + 1}{x} = rac{2x^2 + 1}{x}.$ Since $P(x) ext{ ∪ } Q(x) = 2x^2 + 1$ for $x eq 0,$ we have $(P ext{ ∪ } Q) R (P ext{ ∩ } Q) because $P(x) ext{ ∪ } Q(x) = P(x) ext{ ∩ } Q(x)$ for $x eq 0.$

⚠️ COMMON MISTAKE: Students often confuse the union and intersection of sets, leading to incorrect conclusions about the relation and its properties.