Lecture 2 - Basic Concepts and Proof By Induction Flashcards
What is a sequence of objects?
It is a list of these objects in some order.
Does order matter in a sequence?
Yes! but it does not matter in a set
Does repetition matter in a sequence?
Yes! but it does not matter in a set
What is the power set of a given set A?
It is the set of all subsets of A
What is the Cartesian Product?
The Cartesian Product is the set of all ordered pairs.
What is a function?
A function is something that takes an input and produces an output
What is a predicate (or property)?
It is a function whose range is {TRUE, FALSE}
What is an equivalence relation?
It is a special type of binary relation that captures the notion of 2 object being equal in some feature
What are the 3 conditions of an equivalence relation?
- Reflexive
- Symmetric
- Transitive
What is reflexive?
For every x, xRx
What is symmetric?
For every x y, xRy implies yRx
What is transitive?
For every x y z, xRy and yRz implies xRz
What is a proof?
A proof is a convincing logical argument that a statement is true.
They must be airtight, beyond any doubt
What is proof by induction?
It is a proof that shows that all elements of some infinite set have a specified property.
They have 2 parts, a base case and an induction.
What is the base case?
The first start of proof by induction - prove that P(1) holds
