What is a Relation
Duration: 4 min
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This educational video provides a foundational lecture on the concept of a "Relation" in set theory. The instructor defines a relation as any subset of the Cartesian product of two sets, A and B. He derives a formula for the total number of possible relations based on set cardinalities. The lesson moves from abstract definitions to concrete examples, enumerating all possible relations systematically. Finally, the instructor briefly introduces matrix representation and identifies the largest and smallest possible relations.
Chapters
0:00 – 2:00 00:00-02:00
The instructor starts by defining a relation on the slide: "Let A and B are sets then every possible subset of 'AxB' is called a relation from A to B." He explains that if set A has m elements and set B has n elements, the Cartesian product AxB contains m*n pairs. He draws a diagram with set A={a, b} and set B={1, 2, 3}, drawing red arrows from elements of A to elements of B to visualize the pairs. He writes "m" and "n" above the sets and "mn" above the arrows. He explains that for every single element (pair) in the Cartesian product, there are two choices: either it is present in the subset (relation) or it is not. This binary choice for each of the m*n elements leads to the total number of possible relations being 2 raised to the power of m*n, which he writes to fill the blank on the slide.
2:00 – 4:01 02:00-04:01
To illustrate the concept, the instructor presents a specific example: "For E.g. if A = {a, b}, B = {1, 2}". He lists the Cartesian product AxB = {(a, 1), (a, 2), (b, 1), (b, 2)}. He then displays a large table with four columns corresponding to these four pairs. The rows contain binary values (0 or 1), representing the presence or absence of each pair in a specific relation. This table systematically lists all 16 possible subsets of AxB, demonstrating the 2^4 calculation. He then transitions to a new slide titled "Matrix Representation," showing a grid where rows are labeled x, y, z and columns 1, 2, 3, with cells containing pairs like (x,1). Finally, he states that the "Largest relation possible will be AxB" and leaves a blank for the "Smallest possible relation," which implies the empty set.
The lecture successfully connects the theoretical definition of a relation to practical calculation methods. By defining a relation as a subset of the Cartesian product, the instructor establishes that a relation is essentially a selection of pairs. The derivation of the 2^(m*n) formula is the central mathematical takeaway, highlighting the exponential growth of possible relations as set sizes increase. The detailed enumeration table for the example A={a,b}, B={1,2} serves as a concrete verification of the formula, showing exactly how the 16 combinations arise. The brief introduction to matrix representation and the identification of the largest and smallest relations provide a complete overview of the basic properties of relations, preparing students for more complex topics like equivalence relations or functions.