📘 discrete mathematics

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 $P, Q, R$.

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