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.
