Set Theory Lecture 6 Flashcards
What is mathematical induction?
A proof technique used to establish that a statement holds for all integers greater than or equal to a given base case.
What are the two steps in mathematical induction?
- Base case: Prove the statement holds for the first value. 2. Inductive step: Prove that if it holds for an arbitrary case, it holds for the next case.
What is the formal statement of induction?
Sa ∧ ∀m(m ≥ a → (Sm → Sm+1)) → ∀n(n ≥ a → Sn).
What is an example of an induction proof?
Proving the formula for the sum of powers of 2: ∑k=0^n 2^k = 2^(n+1) - 1.
What is the well-ordering principle?
Every non-empty subset of the natural numbers that has a lower bound has a smallest element.
What does the well-ordering principle imply?
It implies that mathematical induction is a valid proof method.
What is the principle of strong induction?
A form of induction where the inductive step assumes the statement holds for all values up to m to prove it for m+1.
What is an example of a proof using strong induction?
Proving that every integer greater than 1 can be factored into primes.
What is the difference between weak and strong induction?
Weak induction assumes only the previous case, while strong induction assumes all previous cases.
What is a sequence in mathematics?
A function that assigns a value to each natural number.
What is an example of a sequence?
The sequence of even numbers: (0, 2, 4, 6, …).
What is a recursively defined sequence?
A sequence where each term is defined in terms of previous terms.
What is an example of a recursively defined sequence?
The Fibonacci sequence: F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2).
What is the explicit formula for a geometric sequence?
tn = c * r^n, where c is the first term and r is the common ratio.
What is an example of a geometric sequence?
The powers of 2: (1, 2, 4, 8, 16, …).
What is the factorial function?
A recursively defined sequence where n! = n * (n-1)! with base case 0! = 1.
What is the formula for the number of diagonals in an n-gon?
tn = (1/2) * n * (n-3) for n ≥ 3.
What is transitive closure?
The smallest transitive relation that contains a given relation.
What is an example of transitive closure?
If a relation R contains (a,b) and (b,c), then its transitive closure contains (a,c).
What is the notation for transitive closure?
R* represents the transitive closure of relation R.
What is an application of transitive closure?
Finding shortest paths in a graph using the Floyd-Warshall algorithm.
What is the definition of the power of a relation?
Rn is defined recursively as Rn+1 = Rn ◦ R, meaning the composition of R with itself n times.
What is an example of relation composition?
If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R².
What is a recursive definition?
A definition that defines an object in terms of smaller instances of itself.
What is an example of a recursively defined function?
The factorial function: n! = n * (n-1)!.
What is a fundamental property of recursive functions?
They must have a base case to prevent infinite recursion.
What is the relationship between induction and recursion?
Recursion is often proved correct using induction.
What is the importance of induction in computer science?
It is used to prove properties of algorithms, such as correctness and complexity.
What is an example of a proof using induction in computer science?
Proving that the sum of the first n natural numbers is (n(n+1))/2.
What is the significance of transitive closure in database systems?
It is used to compute reachability in directed graphs and relational databases.
