📘 discrete mathematics

1. **Problem statement:** Given sets and relations, solve parts (a) to (f). **(a)(i) Find the composition relation $R \circ S$ where $R: A \to B$, $S: B \to C$:

1. **Show that $t \wedge s$ can be derived from the premises $p \to q$, $q \to \neg r$, $r$, $p \lor (t \wedge s)$.** - From $r$ and $q \to \neg r$, since $r$ is true, $q$ must be

1. Problem (a): Find the power set of $A = \{a,b,c,d,e\}$. The power set is the set of all subsets of $A$.

1. **Problem:** Prove that a connected graph is Eulerian if and only if all its vertices have even degree. **Step 1:** Recall the definition: A graph is Eulerian if it contains a c

1. Discrete mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic. 2. Common topics include logic, set theory, combinatorics, graph

1. The problem asks to identify properties of relation $$R = \{(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)\}$$ on set $$A=\{1,2,3,4\}$$. \nCheck reflexivity: Elements 1, 2, 3,

1. **Solve the recurrence relation** $a_n = 8a_{n-1} - 16a_{n-2}$ for $n\ge 2$ with $a_0=16, a_1=80$. 2. Write the characteristic equation: $$r^2 - 8r + 16 = 0$$

1. Problem (7a): Solve the recurrence relation $$a_n = 8a_{n-1} - 16a_{n-2}$$ for $$n \geq 2$$ with initial conditions $$a_0 = 16$$ and $$a_1 = 80$$. 2. Step 1: Characteristic equa

1. Problem: Solve the recurrence relation $a_n = 8a_{n-1} - 16a_{n-2}$ for $n \ge 2$ with initial conditions $a_0 = 16$ and $a_1 = 80$. Step 1: Write the characteristic equation as