Intro to Relations in Discrete Math 📊

What are relations? We'll be defining mathematical relations with examples - using set theory - in todays lesson!

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We can define relations using sets, and set-builder notation is especially handy! A relation R on a set A is simply a subset of the Cartesian product AxA. Each ordered pair in the relation is thus of the form (x, y) where x and y are in A. The inclusion of (x, y) in R means that x relates to y under the relation R. For example, (2, 3) is in the less than relation on the integers because 2 is less than 3.

We also discuss reflexive, symmetric, and transitive properties of relations - as well as equivalence relations! We also briefly mention how functions are just special types of relations.

SOLUTION TO PRACTICE PROBLEM:

We let A = {0} and R = { (0,0) }. The relation R is in fact an equivalence relation. It is reflexive because every element of A relates to itself. It is symmetric because anytime x relates to y, we also have y relates to x. It is vacuously transitive, because it does not contradict the transitive property. In order to NOT be transitive, a relation must have ordered pairs (x, y) and (y, z) but not (x, z).



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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