Discrete Math

Part B: What is the probability that she will enroll in an algebra course or a biology course?

Part C: Are the two events independent?

Part D: Are the two events mutually exclusive?

If $a$ divides $b$ and $a$ divides $c$, then $a$ divides $b + c$.

If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.

If $a$ divides $c$ and $b$ divides $d$, then $ab$ divides $cd$; verify if this is true.

If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.

If $a$ divides $b$, then $c \times a$ divides $c \times b$.

If $a$ divides $b$ and $a$ divides $c$, then for all integers $x$ and $y$, $a$ divides $bx + cy$.

If $a$ divides 1, then $a = \pm 1$.

Create an adjacency matrix for the given graph G. Identify rows and columns using vertex labels A, B, C, and D. Populate the matrix following these rules: the entry in the i-th row and j-th column is 1 if the vertices represented are adjacent in graph G, otherwise it is 0.

Draw a directed graph that has the following adjacency matrix: $\begin{bmatrix} 0 & 1 & 0 & 2 \\ 2 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}$

Let $R$ be a relation on the set of integers defined by $a \, R \, b$ if and only if $a - b$ is an integer. Prove that $R$ is an equivalence relation.

Let $R$ be the relation on the set of ordered pairs of positive integers such that $(a, b) \, R \, (c, d)$ if and only if $ad = bc$. Show that $R$ is an equivalence relation.

Prove that a given relation $R$ on the set of integers $\mathbb\Z$ is reflexive, symmetric, and transitive.

Determine if each graph has an Euler Path, and if it does, find the Euler Path.

Determine if each graph has an Euler Circuit, and if it does, find the Euler Circuit.

Find an Euler circuit on the given graph.

Suppose that 60% of American adults approve of the way the president is handling his job, and we randomly sample two American adults. Let the random variable X represent the number of those adults that approve. X can take on the values 0, 1, or 2. Calculate the mean and the variance of this probability distribution.

Let X represent the number of heads when this coin is tossed twice. Here's the probability distribution of X. Suppose we want to calculate the expectation of the random variable X.