Discrete Math

Analyze the time complexity of a recursive implementation of the Fibonacci sequence.

Given a sequence generated by the rule $x_n = x_{n-1} + x_{n-2}$, determine the ratio of consecutive terms as it approaches a limit, and prove that this ratio is the golden ratio $\frac{1 + \sqrt{5}}{2}$ or its negative inverse.

We're given a graph of $n$ nodes and an array of edges belonging to this graph. Every edge is undirected and connects two nodes $a$ and $b$. We need to return the number of connected components in the graph.

For all x in the domain D, the predicate P(x) is true.

There exists an X in the domain such that the predicate is true.

For all $x$, there exists a $y$ such that $x - y = 0$. Is this statement true or false?

For all $x$, there exists a $y$ such that $3x - y = 0$. Is this statement true or false?

For all $x$, there exists a $y$ such that $x - 3y = 0$. Is this statement true or false?

There exists an $x$ and there exists a $y$ such that $x + y = 15$. Is this statement true or false?

For all $x$, $x > 5$ implies that for all $y$ such that $y > x$, $y > 5$. Is this statement true or false?

For all $x$, $x > 5$ implies that for all $y$ such that $y > x$, $y > 7$. Is this statement true or false?

Given the sets X = {1, 2, 3, 4, 5} and Y = {a, b, c, d, e}, determine whether a mapping f: X \rightarrow Y is surjective or injective. Show examples for each case.

Find the GCD of 12, 33 using the method shown in the video.

Find the GCD of 25, 150 using the method shown in the video.

Find the GCD of 13, 31 using the method shown in the video.

Find the GCD of 750 and 900 using the Euclidean algorithm.

Find the GCD of 252, 105 using the Euclidean algorithm.

Find the GCD of 1005, 105 using the Euclidean algorithm.

Using Euclid’s Algorithm, find the GCD of 480 and 156.

How can you create a mathematical function that contains all the information about probabilities associated with a dice, such that we can recover all property values from P1 to P6?