Sets and Logic…. Chapters 5 and 6

An experiment….. Sets and Logic…. Chapters 5 and 6

Let A and B be sets, then the cartesian product Order matters! Chapter 4: Relations

Suppose that A and B are sets. We say that any set is a relation from A to B • Examples Relations

Let F be a relation from A to B. F is called a function if for every there is exactly one such that We then write Chapter 5: Functions

Alleged Theorem: Suppose f and g are function from A to B. If then f = g. Ran(f) = Functions

Let then , and for any Composition

Definitions Suppose that We will say that f is one-to-one if We say that f is onto if 5.2 One to One and Onto

Suppose that 𝑓:𝐴→𝐵. 1. f is one-to-one iff 2. f is onto iff One to One and Onto

Let Then f is one to one but not onto. Example

Suppose that is one to one and onto, then Suppose f is a function from A to B and is a function from B to A. Then and . Inverses of Functions

Theorem. Suppose 1. If there is a function , such that then f is one to one 2. If there is a function 𝑔: 𝐵→𝐴, such that then f is onto. More Inverses

Theorem. Suppose 𝑓:𝐴→𝐵. Then the following statements are equivalent 1. f is one to one and onto 2. 3. There is a function such that . Inverses

Theorem: Suppose Then Inverses

Let and define Show that f is one-to-one and onto and find Let Examples

To prove that 1. Prove P(K) 2. Prove for Basic Induction

Prove that for every n, Prove Prove that Example

Let R be a partial order on A. Prove that every finite nonempty set has an R-minimal element. Examples

R is a Partial Order on A if it is reflexive, transitive, and antisymmetric. It is a Total Order on A if it is a partial order and in addition Let

For all if n distinct points on a circle are connected in consecutive order with straight lines, then the interior angles of the resulting polygon add up to Fun and Games with Triangles

Theorem: For any positive integer n, a square grid with any one square removed can be covered with L-shaped tiles. Fun and Games with tiles

Ex: Recursion

Prove that for every . Define a sequence, Find a formula and prove its correctness. Example

For every and every natural number n, Example

Simple Induction… Prove P(k) Prove for n>k, Strong Induction….. Strong Induction

Theorem: For all natural number n and m, if m>0, then there are q and r such that n = mq+r. ( q is the quotient, and r is the remainder ) Examples

Theorem: Every integer n>1 is either a prime or a product of primes. Example

Theorem: Every non-empty set of natural numbers has a smallest element. Theorem: is irrational. I.e. there are no integers p and q such that Example

Sets and Logic…. Chapters 5 and 6