Discrete Math

If you roll a 6-sided die and flip a coin, what is the probability of rolling a 5 and getting heads?

What is the probability of drawing two green marbles without replacement?

Prove that for any integer $n > 4$, $n! > 2^n$.

Prove that $n! > 2^n$ for $n \geq 4$ using mathematical induction.

Determine if a proposition is a tautology, contradiction, or contingency using logical equivalences.

Verify the logical equivalence $\sim (\sim p \land q) \land (p \lor q) = p$ using the laws of logic.

Prove that $1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$ using mathematical induction.

Show that $1 + 2 + \ldots + n = \frac{n(n + 1)}{2}$.

Show that the complement of $A_1 \cup A_2 \cup \ldots \cup A_n$ is equal to the complement of $A_1$ intersect the complement of $A_2$ intersect all the way up to the complement of $A_n$.

Run Prim's algorithm on the given graph to construct a minimum spanning tree (MST).

Run Kruskal's algorithm on a connected graph with weighted edges to find the minimum spanning tree.

What is 53 equivalent to mod 3?

Negate the conditional statement $ ext{P} ightarrow ext{Q}$.

Negate the statement: "If it is raining, then it is cloudy."

Negate the statement: "If it is a toaster, then it is made of gold."

Negate the statement: "If wishes were horses, then beggars would ride."

Negate the statement: "If it is blue then it is not spinach."

Negate the statement: "If q then not p"