Show that Axiom ^4 s ∈𝒫 ℛ ℛ.

Name: Problem 1, Chapter 7: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)
Uploaded: 2020-05-31T19:18:52-08:00
Duration: 3 min 38 s
Channel: Geno Ellis
Description: Show that Axiom ^4 s ∈𝒫 ℛ ℛ.

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Key Concepts

Axioms

Axioms are the basic, unproven assumptions or starting statements in a formal system from which other statements (theorems) are derived. They serve as the foundation of the logical or mathematical framework and ensure that the system has a well-defined basis for deduction.

Formal Systems

Formal systems consist of a formal language, a set of axioms, and rules of inference. They provide a rigorous structure for establishing the truth of statements through systematic derivations, ensuring that proofs are carried out in a precise and unambiguous manner.

Set Membership Notation

Set membership notation, indicated by the symbol '?', is used in mathematics to express that an object is an element of a set. It is a fundamental concept in set theory and logic, providing a clear way to denote relationships between objects and collections.

Axiom Schemas

Axiom schemas are templates that represent an infinite collection of axioms in a formal system. They allow for the generalization of certain properties or rules across various instances, ensuring that the system can accommodate a wide range of specific cases while maintaining a concise foundational structure.

Transcript

00:02 Alright, this time we are given an equal angular triangle and we just want to prove that that's also equilateral, which isn't too too bad it takes a little bit of explaining.

00:19 Okay, so we start with our given we say angle d is congruent to angle e is congruent to angle f and that's given and i just picked the triangle from the actual corollary that this came from, you can pick whatever letters you want.

00:39 Not a big deal.

00:41 Okay, so first step, since the first two given are de and ef, let's start there.

00:48 So how can we show d .e .e .f.

00:50 Are congruent.

00:52 Well, since the angles opposite to them are congruent, we can use the converse of the isocise triangle theorem to show that these two sides need to be congruent.

01:01 So we can say d .e.

01:07 Is congruent to e .f.

01:13 And this is the converse, isosceles triangle theorem.

01:22 Let's number our steps just in case.

01:26 So step two, this is step one.

01:30 Okay.

01:31 And then we can do that again for ef and fd.

01:36 Especially if we could, well, let's pick blue.

01:38 I like blue.

01:39 So we say ef and fd.

01:45 So these two have to be the same for the exact same reason.

01:48 Conversi sossi's triang there.