📘 discrete mathematics

1. **Problem Statement:** Given a positive integer $N$, find the starting number $\leq N$ that produces the longest Collatz sequence chain. If multiple numbers have the same longes

1. The problem asks to identify the given sets and ordered pairs. 2. The first given is \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, which is a set of numbers from 1 to 9. A set is a collection

1. مسئله ۷۷: تعداد اعضای زیرمجموعه‌های مجموعه $A = \{1, 2, 3, ..., 30\}$ را بیابید. هر زیرمجموعه می‌تواند شامل هر تعداد عضو از ۰ تا ۳۰ عضو باشد. تعداد کل زیرمجموعه‌های یک مجموعه با

1. Let's clarify the problem: Discrete mathematics covers topics like logic, set theory, combinatorics, graph theory, and algorithms. 2. Since no specific question was given, pleas

1. **State the problem:** We have two sets:

1. The problem asks to find the rule for how the minimum number of vertices with match heads increases as the number of house figures increases. 2. From the description, the first

1. **Problem Statement:** Explain what an equivalence relation is and demonstrate its use with an example. 2. **Definition:** An equivalence relation on a set is a relation that is

1. **Problem statement:** We are given two graphs represented by their adjacency matrices and need to (a) find the number of paths of length 3 between each pair of vertices and (b)

1. We start with the relation $R = \{(1,2), (1,3), (2,4), (3,2), (4,3)\}$ on the set $A = \{1, 2, 3, 4\}$. We represent it as an adjacency matrix $M$, where rows and columns corres

1. **State the problem:** We need to show that the relation $R = \{(a,b) \mid a \leq b\}$ defined on the set $S = \{2,4,6,8,10\}$ is a partial order relation. 2. **Definition of pa

1. The problem is to determine if the given relation \( f \) from set \( D = \{1, 2, 3, 4\} \) to set \( Y = \{a, b, c, d\} \) defined by: \( f(1) = a, f(2) = b, f(3) = b, f(4) = c

1. **Problem statement:** Given the relation $R$ on $\mathbb{N}$ defined by $xRy \iff \frac{2x + y}{3} \in \mathbb{N}$, check pairs, prove $R$ is an equivalence relation, and find

1. The problem asks us to draw the Hasse diagram for the partially ordered set $(\{1, 2, 3, 4, 6, 8, 12\}, \mid)$ where the order relation is divisibility: $a \leq b$ if and only i

1. **State the problem:** We are asked to draw the Hasse diagram for the partial order defined by the divisibility relation $a \mid b$ on the set $\{1, 2, 3, 4, 6, 8, 12\}$.

1. **Problem 1: Number Systems in Computing** a) The binary, octal, decimal, and hexadecimal number systems are fundamental in computing because they uniquely suit digital electron

1. Given sets and relations: \(R = \{(1,2),(1,3),(2,3),(3,1),(3,3)\}\)

1. Stating the problem: We have relations $R$ and $S$ on the set $\{1,2,3\}$: $$R=\{(1,2),(1,3),(2,3),(3,1),(3,3)\}$$

1. Consider the problem involving set operations and with conditions: Given:

1. The problem is to find the power set of a given set, but the set itself was not specified in your question. 2. The power set of any set $S$ is the set of all possible subsets of

1. The problem asks to state the Boolean expression laws. 2. Boolean algebra has several fundamental laws that govern the operations AND ($\cdot$), OR ($+$), and NOT ($\overline{\c

1. **Construct a truth table for the compound proposition $(P \to R) \land (Q \lor \neg R)$**. - List all possible truth values for $P, Q, R$.