Complement of a Relation
Duration: 3 min
This video lesson is available to enrolled students.
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An AI-generated summary of this video lecture.
The video lecture focuses on the mathematical concept of the complement of a relation within set theory. The instructor defines the complement of a relation R from set A to set B. He explains that the complement, denoted as R', R^c, or R_bar, consists of all ordered pairs present in the Cartesian product A×B that are not present in the original relation R. The lecture transitions into a practical example. By subtracting the elements of R from the total Cartesian product, the instructor demonstrates how to find the complement R'. Finally, the video concludes by establishing two fundamental properties: the union of a relation and its complement equals the universal set (A×B), and their intersection results in an empty set.
Chapters
0:00 – 2:00 00:00-02:00
In this segment, the instructor introduces the definition of the complement of a relation. The slide displays the text "Complement of a relation: - Let R be a relation from A to B, then the complement of relation will be denoted by R', R^c or R_bar." He explains the set builder notation and the set difference formula. Visually, he underlines key terms like "Complement of a relation" in red. He also writes the alternative notations R', R^c, and R_bar in red ink on the right side. He emphasizes the condition (a, b) ∉ R by underlining it, clarifying that the complement contains pairs not in R. He further underlines (A×B) in the subtraction formula to highlight the universal set for the relation.
2:00 – 2:39 02:00-02:39
The instructor moves to a worked example. The slide shows A×B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} and R = {(a, 1), (a, 3), (b, 2)}. He crosses out elements of R within A×B to identify the complement. He writes the result R' = {(a, 2), (b, 1), (b, 3)}. Following the example, he introduces two properties: R U R' = and R ∩ R' =. He writes the answers in red ink, stating that the union R U R' equals A×B and the intersection R ∩ R' equals the empty set symbol Φ.
The lesson systematically builds understanding from definition to application. It starts with the formal definition of the complement of a relation using set notation and set difference. The instructor clarifies that the complement is essentially the "rest" of the Cartesian product after removing the relation itself. This theoretical foundation is immediately reinforced with a numerical example where specific ordered pairs are listed, and the complement is derived by exclusion. The lecture concludes by solidifying the relationship through set operations. These final properties confirm that the relation and its complement are disjoint and their union covers the entire domain of possible pairs, ensuring a complete partition of the set.