Discrete Math

For a given function $f(n) = 2n + 3$, determine the Big O, Omega, and Theta notation expressions.

Prove that a specific algorithm has an upper bound according to Big-O notation.

Prove by induction that the sum of the series $\sum_{r=1}^{n} r^2$ is equal to $\frac{n(n+1)(2n+1)}{6}$.

Prove that the sum of the squares of the first $n$ natural numbers can be expressed as: $1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$ for all natural numbers $n \geq 1$ using proof by induction.

Prove that the intersection of $A$ and $B$ is a subset of $A$ union $B$.

Prove that $A - B \cap C$ is a subset of $A - B \cup A - C$.

Prove De Morgan's Law: $(A \cup B)' = A' \cap B'$.

Prove that if $A$ is a subset of $B$, then $A - C$ is a subset of $B - C$.

Write an equivalent logical expression using quantifiers for the statement: "A union B is a subset of C difference D".

Write an equivalent logical expression using quantifiers for the statement: "A union B is not a subset of C difference D" using the negation of previous statements.

Define the nth number $a_n$ in a Fibonacci sequence such that $a_n = a_{n-1} + a_{n-2}$ for $n \geq 2$, with initial conditions $a_0 = 0$ and $a_1 = 1$.

Define the recurrence relation for a geometric sequence as $a_n = a_{n-1} \times 2$ for $n \geq 1$, starting with $a_0 = 3$.

For the sequence defined by $0, 2, 6, 12, 20, 30, 42$, show that $a_n - a_{n-1} = 2n$ and prove that $a_n = n(n+1)$ given $a_0 = 0$.

Solve the second-order linear homogeneous recurrence relation $a_n = 5a_{n-1} - 6a_{n-2}$ with initial conditions.

Given the recursive formula $a_{n+1} = 3a_n + 2$, and the first term $a_1 = 1$, find the next four terms.

Given the recursive formula $a_n = 2(a_{n-1})^2 - 5$, with the first term $a_1 = 2$, find the next three terms.

Given the recursive sequence defined by $a_0 = 1$ and $a_n = 1 + a_{n-1}^2$, compute the first few terms of the sequence.

Using the recursive relation for a sequence similar to the Fibonacci sequence, where $a_0 = 1$, $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2} + a_{n-3}$, find the first few terms of the sequence.

Using the logistic sequence recursive relation $a_{n+1} = 2 \cdot a_n \cdot (1 - a_n)$, with an initial term between 0 and 1, calculate the behavior of the sequence.

Given a set with elements $A, B, C, D$, determine if the relation is reflexive, symmetric, and transitive by considering the arrows between the elements.