Composition of Relations (ZFC Set Theory Lecture 2.11)

In school you probably learned how to find the composition f∘g of two functions, but did you know we can generalize this to relations? If R and S are binary relations, we define their composition R∘S = {(x,z) : (x,y)∈S and (y,z)∈R for some y}. Intuitively, we take ordered pairs from S and try to "glue" them to ordered pairs in R. ZFC Set Theory is super important to learn because it forms the foundation of (almost) all modern mathematics. Algebra, analysis, topology, you name it, all rely on Set Theory. All the results of these diverse branches of math can be, at least in principle, described in the language of sets and proven from the ten axioms of Set Theory. If you want to learn more about this fascinating branch of mathematics, watch my ZFC Set Theory Course (ideally in order):    • ZFC Set Theory Course   CHAPTERS: 0:00 Introduction 0:31 Definition of R∘S 1:06 Proof of existence 2:02 Practice computing R∘S and S∘R 5:36 Exercise: R∘∅ = ∅ and ∅∘R = ∅ 6:48 Theorem: R ∘ id𝐴 = {(x,y)∈R : x∈A} 9:42 Theorem: id𝐴 ∘ R = {(x,y)∈R : y∈A} 12:02 Why R⁻¹ ∘ R is not always identity 14:26 Theorem: R⁻¹ ∘ R ⊇ id_dom(R) 16:19 Theorem: R ∘ R⁻¹ ⊇ id_range(R) 18:12 Theorem: (R∘S)[A] = R[S[A]] Please like and subscribe if this video was helpful. Your support means a lot to me. Got any questions? Ask them in the comments, and I'll try to answer them. Also, I'm available for private online math tutoring via Zoom, so message me if you would like to schedule an appointment.