📘 discrete mathematics

1. **Problem Statement:** Define a graph in the context of discrete mathematics and graph theory, and provide several numerical examples. 2. **Definition:** A graph $G$ is a pair $

1. **Problem Statement:** Find the period of the multiplicative linear congruential generator (MLCG) defined by the recurrence relation $$x_{n+1} = (a \times x_n) \bmod m$$ with pa

1. **State the problem:** We are given a mapping from a set of inputs to outputs as follows: 2 maps to 8, 8 maps to 2, 5 maps to 5, and 7 has no output. 2. **Identify the type of r

1. **Problem statement:** We have a recurrence relation defined as $$a_n = a_{n-1} + a_{n-3}$$ for $$n \geq 3$$ with initial conditions $$a_0 = 1$$, $$a_1 = 1$$, and $$a_2 = 1$$. 2

1. **Problem Statement:** Given the set $A = \{2, 3, 4, 5\}$, define a relation $T$ on $A$ such that for every $x, y \in A$, $x T y$ if and only if $4 \mid (x + y)$, meaning $x + y

1. **State the problem:** We want to solve the recurrence relation $$a_n - 2a_{n-1} - 3a_{n-2} = 0$$ for $$n \geq 2$$ with initial conditions $$a_0 = 3$$ and $$a_1 = 1$$ using gene

1. The problem asks us to implement the Collatz function $f(n)$, which is defined as: $$

1. **Problem 1: List all subsets of the set {0, 5} and find the number of subsets.** - The set is $\{0, 5\}$ which has 2 elements.

1. **Problem:** Given a survey of 25 cars with options air-conditioning (A), radio (R), and power windows (W), find various counts of cars with specific option combinations. **Step

1. **Discuss all features of a graph:** A graph consists of vertices (nodes) and edges (connections between nodes). Key features include:

1. **Problem 1: Construct the truth table for the biconditional statement:** "It is raining if and only if the ground is wet." The biconditional statement $p \leftrightarrow q$ is

1. The problem asks for a Hasse diagram, which is a graphical representation of a finite partially ordered set (poset). 2. A Hasse diagram shows the elements as vertices and the or

1. The problem is to understand and create a Hasse diagram, which is a graphical representation of a finite partially ordered set (poset). 2. A Hasse diagram shows elements as vert

1. **Problem (a): Analyze parallel and serialized tasks from a Hasse diagram** Given a Hasse diagram representing a partially ordered set (POSET) of tasks, we identify which tasks

1. **Problem statement:** Solve the difference equation $$y_{n+2} - 7y_{n+1} + 12y_n = 2^n$$ with initial conditions $$y_0 = 0$$ and $$y_1 = 0$$ using the Z-transform. 2. **Recall

1. **Statement of the problem:** Prove by induction that for all natural numbers $n$, the expression $$4^n + 6n - 1$$ is divisible by 9. 2. **Formula and induction principle:** We

1. Problem 1: Definitions, language concatenations and a counting problem from the exam paper. 1. (a) Definitions.

1. Задатак 10: Дати су елементи и табеларни приказ релације ρ. Табела показује да су у релацији парови (1,1), (2,1), (b,b) и (a,a).

1. Define two elements Boolean algebra: Boolean algebra is a mathematical structure consisting of a set with two elements, usually 0 and 1, along with two binary operations (AND \(

1. Problem: Give an example of a simple graph with 12 vertices and 35 edges. Step 1: Recall that a simple graph with $n$ vertices can have at most $\frac{n(n-1)}{2}$ edges.

1. **Stating the problem:** We need to find the maximal and minimal elements of the poset (partially ordered set) defined on the set $\{3,5,9,15,24,45\}$. 2. **Understanding the po