Discrete Math

Verify if the given relation is transitive by checking all possible paths between elements.

Given a set $A$ with elements, determine if the relation on $A$ is reflexive, symmetric, or transitive based on the arrow diagram provided.

Show that T is true from the following premises using rules of inference.

Prove $Q \land R$ given $P \implies Q \land R$ and $P$.

Prove $T$ is true given that $P \land Q$ is true, $P \lor S$ or $P \lor S \implies \lnot R$ is true, and $R \lor T$ is true.

What is the intersection of set A and set B, where set A contains the elements \{2, 4, 5, 6, 9\} and set B contains the elements \{2, 3, 5, 6, 7, 9, 10\}?

What is the intersection of sets C and D, where set C contains {3, 4, 6, 7, 10} and set D contains {3, 6, 8, 9}?

Determine the intersection of sets F and G where set F contains \{a, b, c, d, f, g, j\} and set G contains \{a, c, g, h, k\}.

Determine the intersection of sets J and K, where set J contains \{5, 7, 10, 11\} and set K contains \{2, 4, 8, 13\}.

What is the intersection of set R \{3, 4, 7, 10\} and an empty set S?

Find the union of sets A and B, where set A contains \{1, 2, 3, 4\} and set B contains \{3, 4, 5, 6\}.

Find the union of sets C and D, where set C contains {3, 5, 9, 11, 13} and set D contains {2, 3, 6, 8, 12}.

Determine the union of sets J and K, where set J contains {a, c, d, e} and set K contains {a, b, f, e, g}.

What is the union of set X $\{2, 5, 8, 12\}$ and an empty set Y?

Determine the intersection and union of sets A and B, where set A contains {3, 4, 5, 7} and set B contains {2, 4, 5, 8}, and represent the results using a Venn diagram.

Given a linear homogeneous recurrence relation with initial conditions, solve the recurrence relation using the characteristic equation technique. The example involves getting all terms over to the left side to form the characteristic equation, finding the general form of the solution, and determining constants using initial conditions.

Solve the recurrence relation $T(n) = 2T(n/2) + 4n$ with the initial condition $T(1) = 4$ using the iterative substitution method.

Solve the linear congruence: $5x \equiv 2 \pmod{9}$.

Solve $17x \equiv 3 \pmod{29}$ using Euclid's Algorithm.

Prove that for the recursively defined sequence where the first term is 1, the second term is 3, and the k-th term is defined as $a_k = a_{k-2} + 2a_{k-1}$, all terms are odd.